On a Thursday in mid-January, an ice storm that will go on to create tornadoes in Texas and a 20-car pile-up in Kansas is stopping off first in California, though admittedly in less apocalyptic fashion. Los Angeles is not a town designed for rain: the candy-coloured buildings seem somehow both drab and gaudy, and as the water drips from their leaves the palms look suicidal.

Standing at the window of a tower block in Burbank, a few miles northeast of Hollywood, surveying the office buildings below and beyond them, the hills, Jake Gyllenhaal sees things differently.

"I think there's something kind of magical about the way the palm trees come out of the mist," he says.

And he's right. It's true. There is. Perhaps the 36-year-old actor is just pleased to see daylight. Before our interview, he has been holed up in an edit suite with David Gordon Green, the director of Pineapple Express, who is now directing Stronger, the first film being made by Gyllenhaal's production company, Nine Stories. As well as producing, Gyllenhaal is playing the lead role of Jeff Bauman, who lost his legs at the 2013 Boston Marathon bombing and upon whose book the film is based.

The pair emerge laughing when a production assistant announces my arrival. Gyllenhaal — in bearded mode for a future part — introduces himself. "Hi, I'm Jake." Yes, sure enough, there are those famously intense blue eyes, the heavy eyebrows, and the crescents in his cheeks when he smiles. He looks his age, though the weathering is making his face less pretty but more handsome, more interesting, in that way in which some men luck out, and most do not.

Gyllenhaal excuses himself for a moment and Green, when Gyllenhaal is out of earshot, expresses what seems like genuine enthusiasm about working with the actor-now-producer in these close quarters where, he says, he can "milk him for his weirdness". It's a strange way of putting it: as though "weirdness" is a precious commodity to be bottled and processed like anti-venom. But then you think about Gyllenhaal's career and the roles he's best known for, and how they fit into a Hollywood spectrum of endless superhero franchises and vapid fantasy epics and money-spinning sequels, and it makes a certain sense.

""Yeah, have some restraint," he quips.

There's Donnie Darko, the 2001 cult film about an over-medicated teenager who sees an imaginary six-foot rabbit that predicts end times, and which still holds up as an evocative mood piece about the terrifying, numbing potentiality of adolescence. There's Jarhead, Sam Mendes' film about a Marine in the first Gulf War, which revels not in heroism or danger but in the existential void the war created — of purpose, of meaning, of fun. There's Ang Lee's 2005 drama Brokeback Mountain, in which Gyllenhaal and the late Heath Ledger played cowboys who stumble into an intimate relationship that surprises and overwhelms them both, and was a film which changed the way homosexual love was portrayed on screen and how it was talked about after.

More recently there was Nightcrawler, a searing satire on the amoral appetites of the news media (but not other kinds of media, oh no!) told through the actions of a feral, sociopathic cameraman. Last year, there was Nocturnal Animals, Tom Ford's seductive and savage drama about the damage people do to each other, in body and mind, in which Gyllenhaal played a devastated lover, Edward, but also Tony, a character in a book Edward writes, whose own sense of identity — as a father, as a husband, as a protector — is shattered in the most brutal of ways.

So yes. It looks like David Gordon Green isn't the only one to milk him for his weirdness. Which is also, in a manner of speaking, what I'm supposed to be doing today in Burbank. But first we have to work out where to sit.

We're in a small room next to the suite in which he and Green are working; Gyllenhaal's publicist has already phoned to apologise for the "lack of atmosphere" and to dash my hopes we'll conduct the interview while riding BMXs down Santa Monica beach, or whatever it is Angelenos do in their leisure time. Mysteriously, the space is disproportionately full of furniture, making it feel like we're at an improv workshop full of scene prompts: there's a single black leather office chair tucked under a wooden desk ("Dispiriting Job Interview at Accountancy Firm"), two striped armchairs side by side beneath the window ("Waiting Area of Suburban Beauty Parlour") and a dark leather couch against the wall ("Introductory Session with Cut-Price Psychiatrist"). The latter seems most apt.

"You take the couch," Gyllenhaal says, pulling up an armchair. He's supposed to be "The Patient" in this scenario, I remind him.

"Oh, right," he says.

But that seems odd, too. Eventually, we both sit on the couch. I apologise for the intimacy, given the plethora of seating options available.

"Yeah, have some restraint," he quips.

I vow to do my best.

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Journalists often say interviewing Jake Gyllenhaal is a little tricky. Not just because he's careful about his privacy, which he is, but also because, as he says, leaning back into the couch, "I have an abstract mind... I can't do small talk, it's not an easy thing for me. I think I tend to ostracise myself as a result of that." He pauses. "I think growing a little older, I'm totally fine with that."

Also, he likes to turn ideas round and round, inside out and upside down. He does it with journalists, directors, fellow actors, testing the material, pushing, probing; reputedly sparing no feelings if he thinks it serves the common good. He's inquisitive to a fault and sometimes his interest in other people's experience only makes his own more opaque; in one recent cover story with a British style magazine for which he was interviewed by Tom Ford, it was the director, and not the actor, about whom one learned the most. (At one point he asks me what it's like doing my job and I give him my tuppence-worth about the hideousness of hotel junkets for a good couple of minutes before I remember that, oh yes, no one cares.)

He can certainly riff on the stupid questions that make silly quotes to be circulated on the internet for infinity — a case in point: the time he made an off-hand comment to a radio station about his fondness for baguettes from a certain British high-street bakery; even The Guardian ran a story headlined "Jake Gyllenhaal: I love Greggs" — but that's not to say he won't look at you with a certain quizzical resignation when you ask them. Also: just you try looking into those baby-blues for an hour-and-a-half. It's no mean feat.

"I think I sometimes push things too far"

Today's encounter is made just that little bit more abstract by the fact we are nominally here to discuss a film, Life, which neither of us has seen. Life appears to be a good old-fashioned psychological thriller set in the International Space Station, where a group of preternaturally attractive astronauts and one cosmonaut discover the rover they have sent to Mars has brought back a passenger. "A cellular being," says Gyllenhaal. "They're really excited about it, and all of a sudden it becomes a terrifying interaction with this creature."

And you get picked off one by one?

"Yeah, I don't know!" He says. Though, of course, he does.

The film was made at Shepperton Studios in Surrey (during which time Gyllenhaal lived "in a house in Notting Hill") and to recreate the appearance of zero gravity, the actors were suspended from wires, Peter Pan-style. "It was very fun to do physically," he says. "In terms of the emotional world, in terms of conjuring feelings up while floating on wires, it proved to be a little more difficult than I thought it was."

The director, Daniel Espinosa, also wanted the actors to stick together as much as possible, to recreate some sense of life on the International Space Station: the camaraderie, the irritation, the claustrophobia, the isolation. "We were pretty much on wires all day long, and then when we were off the wires there were tents off-stage that we would all go into. We rarely went outside. There was this 'flu that passed around everybody, that just rapidly took us all down."

It sounds like a hoot. It also sounds like the kind of self-punishing role that Gyllenhaal seems to have a jones for. Like Southpaw, the 2015 movie in which Gyllenhaal bulked up and learned to box in order to play fictional world light-heavyweight champion Billy "The Great" Hope. Or Everest, an account of a disastrous 1996 expedition, in which Gyllenhaal, as doomed mountain guide Scott Fischer, had to enact being frozen to death in the snow which, given they were filming at an altitude of 4,000ft in temperatures of –30°C, probably wasn't too difficult to do. Or Nightcrawler, for which Gyllenhaal, of his own volition, shed 30lbs to give his character, Lou Bloom, a haunted, hungry look, by running 15kms a day and eating kale. ("I learned that that was a really bad thing to do by the way," he laughs.)

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It's a reputation Gyllenhaal is well aware of. "People always say, 'I've heard you're very committed to your roles, you've lost weight', which seems to be some, like, magical, extraordinary thing," he hastily adds a diplomatic caveat, "though I know it's not easy for a lot of people in the world… People just love talking about the physical part of it. 'What did you do to get in shape for this thing? What did you eat? What was your diet?'"

Even his peers have been known to probe him. "Some actors have asked me, 'Hey, I know when you get into something you do a lot of preparation; I'm doing this role, do you think I should go and jump off a building?' I'm like, 'No! Use your imagination!'"

But it's also a reputation for which, he's aware, he's partly got himself to blame. "I think I sometimes push things too far, and I realised that I didn't really get the result that I wanted because it wasn't always fun, you know?" Life, he says, was appealing because "we were literally going to just fly", and the process did indeed prove, he says, to be "a bit less torture. Ha!"

"I love the idiosyncrasies of people"

"I've grown up thinking somewhere that if you're really doing something great it has to be punishing in some sort of way," he says. "I've moved from that. After a few years of really pushing in different areas, pushing my body physically, pushing my mind, going a little too far in spaces, I've realised that joy is a huge part of it."

Did he feel playing all these souls in crisis was doing him long-term damage? "I don't know. I think, fortunately or unfortunately, I'm closer to the characters I play than I'd like to think. There are probably people who, if they heard me say that, wouldn't want to hang out with me…" He trails off.

Did anyone try to convince him to ease up on the self-flagellation?

"Yeah. My mom. All the time."

Gyllenhaal has had longer than most to make peace with his profession. He was born in Los Angeles to a screenwriter mother, Naomi Foner (who wrote Running on Empty, for which girls who grew up in the Eighties — holler! — will be eternally grateful), and a director father, Stephen Gyllenhaal; his older sister is the actress Maggie Gyllenhaal. His first screen role, at the age of 10, was as Billy Crystal's smart-mouthed, spike-haired son in City Slickers, and though he attended regular high school — regularish, it was the prestigious Harvard-Westlake — and then college, studying Tibetan Buddhism and eastern mysticism at Columbia, he left after two years to pursue acting.

He was drawn to it, right from the beginning, because "it felt like I could take something and have an idea and study those words and see if there was a place where it triggered something in me… I like the sound and the feeling of words coming from my mouth. I love the idiosyncrasies of people and what they're trying to communicate unconsciously through their behaviour. I'm always searching for honesty."

Really? When you were, like, 12?

"Mm-hmm."

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As well as the good stuff, he's done a couple of duds, most notably Prince of Persia: the Sands of Time, a Disney/Jerry Bruckheimer video-game-adaptation mega-production that, it's fair to say, did not excite the critics (The New Yorker: "Schlock"; The New York Times: "Mush"). It's often stated that that movie was a sort of epiphany for Gyllenhaal, provoking a seismic change that led him to cast off cynical Hollywood behemoths in favour of weirder, smaller options. Naturally, as with most attempts to smooth the edges, he resists. "There's so many seismic changes: everything responds to everything else. People like to talk about what they can quantify, what they consider unsuccessful as opposed to something that is, but I just don't look at it like that."

As to whether or not he'll enter the blockbuster fray ever again, he's noncommittal. "Maybe, no. I don't know. There's no orthodoxy. I don't have a resentment or a need to do one thing or another thing."

So now he's in his mid-thirties, he's been an actor for a good quarter-century, and has had plenty of time to get used to negotiating the peculiar life he has chosen. He's talked often about how "absurd" acting is, but surely today it is weirder than ever, where every facial expression becomes a gif, a meme, a puff of online gossip, and where every action is scrutinised. He mentions in passing that he's just read David McCullough's Pulitzer Prize-winning biography of the second President of the United States, John Adams, while on Christmas vacation; the Mail Online reported on this very trip, but chose not to focus on his love of non-fiction and instead ran paparazzi shots of him in his swimming trunks under the headline, "Jake Gyllenhaal shirtless as he enjoys holiday in St Barths".

But on the other hand, actors have also got more power, or at least, thanks to proliferating platforms, more opportunities to relay their opinions should they wish to. And these are weird times. The weekend before our interview, Gyllenhaal presented an award at The Golden Globes, where Meryl Streep made her widely circulated speech decrying the bigotry of President Donald Trump, and the then-incumbent president responded, in a tweet, that she was "one of the most overrated actresses in Hollywood". In his speech at the same awards ceremony, the British actor Tom Hiddleston seemed to imply that the show for which he was collecting a statuette, The Night Manager, had eased the suffering of Médecins Sans Frontières staff in South Sudan. So flat did it fall that he had to issue an apology. Actors are revered and reviled more than ever before.

"I'm not very interested in what I think because I'm a wealthy white male"

Gyllenhaal says he feels no particular duty to use his fame for grand messages of his own — "I'd love to use whatever audience I have to make sure those people who know what they're talking about are heard" — though his actions suggest he's relatively politically engaged; his family have actively supported the American Civil Liberties Union, he took part in a Broadway fundraiser for Hillary Clinton and a week after this interview he and sister Maggie attended the Women's March in Washington DC.

When we meet, it is just over a week until Trump's inauguration; on the side of a building on Sunset Boulevard, a gigantic picture of the political comedian Bill Maher looms, like the eyes of Dr TJ Eckleburg, beneath the words: "Make America Sane Again." It's almost impossible not to talk about how things are in America, but also Britain and Europe, which he does, knowledgeably, though after a little while he stops me short.

"I'm not very interested in what I think because I'm a wealthy white male," he says, "and right now what's important is not what I think. But I will say I don't like it when people say, 'You shouldn't hear what an actor has to say about something, let the politicians talk', because the irony is that we have elected a celebrity as a president, so as a result that doesn't work any more."

A few years ago, when Gyllenhaal found himself in Washington DC, he asked some people to ask some people about going on a tour of the White House. He received some hints back from some people through some people that he might want to wear something more formal. "I thought that we were maybe going to run into the vice-president or something, so it was like, 'Oh my God!' I went and bought a new pair of shoes." While on the tour he was whisked to one side and into the Oval Office where President Barack Obama was waiting for him.

"It was an amazing feeling, but not a comfortable one," he says. "I was just, sort of, not in my body."

And who said what to whom?

"I don't really remember much of it, but I do remember him saying that, at the time, 'Our country is in a really hard place and there are many people struggling, and it's your job to entertain us, and to bring some levity or joy, or to move us.' I loved that he said that. 'Just as I would give anybody else an objective, this is yours. This is my request of you.' And it was like, 'OK.'"

"It was an amazing feeling, but not a comfortable one"

In keeping with President Obama's command, and assuming Barack enjoys a show as much as the next guy, a few days after our interview, Gyllenhaal is set to fly to New York, where his mother and sister live and he now lives also, with his German Shepherd, Leo, to start rehearsing a musical, Stephen Sondheim's Sunday in the Park with George, which will be performed at the Hudson Theatre on Broadway. Gyllenhaal's talent for singing might not have been called for in his movies — there weren't a lot of jazz hands in Jarhead — but his family is musical, he says, and theatregoers in New York have already had the chance to see him in Little Shop of Horrors and a stripped-back, concert version of Sunday in the Park with George, both of which received glowing reviews. (If you're really curious to check out his pipes, you can also find YouTube clips of him singing "And I am Telling You I'm Not Going" from Dreamgirls on Saturday Night Live — dressed in a sparkly cocktail dress.)

Despite his professed plans to lighten up, Gyllenhaal has chosen a very Gyllenhaalian musical to get his teeth into. Sunday in the Park with George is inspired by 19th-century artist Georges Seurat's painting "A Sunday Afternoon on the Island of La Grande Jatte" and considers the circumstances in which it was painted, and the legacy of Seurat's descendants and also, oh why the hell not, the meaning of art. It is also, by any show-tunes standard, incredibly difficult to sing. When he was offered it, his reaction was typical. "I thought, 'Shit. That scares me'. So I figured if it terrified me enough I should probably do it. And that character, for some reason, feels so close to me."

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Sunday in the Park with George is partly about Seurat's relationship with his lover, Dot, and their inability to truly connect to each other, despite their mutual affection, partly because of George's unswerving devotion to his work. My inner cut-price psychiatrist stirs. Close to you, you say?

"Yes. Ha! Yes and no," says Gyllenhaal. "I think he has a big heart, and doesn't necessarily know how to communicate it with words, but I think actually the missing piece of that show that I found is how much love he has, and how much he has to give, and how signals just get crossed… And so to me, yes he is, he's obsessed with his work, he loves it deeply, it's a part of him. It keeps him from some connection but it also connects him deeply, and all he wants is for someone to see him just as she wants the same thing, and so often things like that happen in our lives."

And just as I'm trying to digest that, and work out how much he's talking about George, and how much he's talking about Jake, he starts to elaborate, kind of, by speaking lines from the play, until he's enacting a conversation between George and Dot in which they try to grasp for each other with words but fall inexorably short.

"I can only do it by singing"

"There are so many things in it, and it's so beautiful. She says, 'What you care for is yourself,' and he says, 'I care about this painting, and you will be in this painting.' You know? And she says, 'I thought you understood,' and he says, 'There's nothing I can say, is there?' And she says, 'Yes George, you can tell me not to go.' She says it to me: 'Tell me not to go…'"

And it's incredibly actory, and you could skewer him for it if you wanted, but it's also kind of a treat, because he's good at this stuff, and you remember that in a few weeks' time Broadway audiences will be throwing roses at his feet, or whatever it is New Yorkers do at the theatre, and it seems as though the house lights really do dim just a little in the weird little improv-workshop junk room.

Then his phone bleeps and our time is up. Gyllenhaal stands to go but keeps reciting, enjoying the sound and feeling of words coming from his mouth, and you get a sense of what The New York Times' reviewer of the concert version of Sunday in the Park meant when he described "this rushing realisation, both lonely and consoling, that looking — and really seeing — another person may be the highest form of intimacy there is." And then he gets to a bit where he's not sure of the words and he says, "I can only do it by singing," and you think, well that's that then, but then he does start singing, or half-singing anyway, looking you in the eye.

"Look at all the things you gave to me, things I hadn't looked at till now. Flower in your hat. And your smile. And the colour of your hair. And the way you catch the light. And the care. And the feeling. And the life."

And then he suddenly snaps out of it. "That's the way it ended. Ha!"

And he's right. It's true. It was.

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Life is out in cinemas on 24 March

Photographs by David Slijper, Styling by Jeanne Yang

Lettermark
Miranda Collinge
Deputy Editor

Miranda Collinge is the Deputy Editor of Esquire, overseeing editorial commissioning for the brand. With a background in arts and entertainment journalism, she also writes widely herself, on topics ranging from Instagram fish to psychedelic supper clubs, and has written numerous cover profiles for the magazine including Cillian Murphy, Rami Malek and Tom Hardy.