One of the most important things to me is risk,” says Jake Gyllenhaal, when we meet amid the hustle and bustle of the Cannes film festival. “I think a lot of people try to stay within some sort of comfort zone and it’s never something that I feel comfortable with, be that subject matter or be that choice in character.”

He isn’t kidding. From the teen angst and time travel movie Donnie Darko, through the Oscar-winning gay Western Brokeback Mountain, to the chilling capitalism horror story Nightcrawler, and idiosyncratic rumination on grief Demolition, Gyllenhaal, has always shown a taste for the off-centre.

He has done Hollywood blockbusters, but the video game adaptation Prince of Persia: Sands of Time, and climate change disaster movie, The Day After Tomorrow now feel like anomalies. Considering what he’s looking for, this isn’t surprising.

“When I read a script, I want it to feel new, like you’ve never seen it before,” he says with genuine passion.

Keeping things fresh means being prepared to fail and you feel that for Gyllenhaal, film-making and acting are sometimes like an extreme sport, where the aim is to challenge himself, and audience perceptions. He couldn’t box when he signed on for the boxer-on-his-uppers tale Southpaw, for example. But time spent in the gym, and living the life of a pugilist for months, turned him into a convincing fighter.

His latest role, as a nebbishy TV wildlife show host, Dr Johnny Wilcox, in cult South Korean film-maker Bong Joon-ho’s brilliant creature feature, Okja, is arguably his most bizarre so far. Doing a kind of Steve Irwin/Groucho Marx/Woody Allen/Sasha Baron Cohen mash-up, Gyllenhaal throws caution to the wind and goes big, brash and camp in a way that could well cause Marmite-like controversy with viewers. Call it his Jack Sparrow moment.

“The wonderful thing about Bong is that he wanted to see and do something totally different,” he says. “Something sort of outlandish. And I love that.”

His performance will cause raised eyebrows among those who look to Gyllenhaal for delicately nuanced work (of which there are still flashes in Okja), but that’s part of the fun, he says. “I’ve talked to people about my performances before and I was like, ‘You know, there’s a subtlety to what I do so sometimes you don’t even notice what I’m doing.’ So I was like, ‘f*** it. Let’s try something totally the opposite.’” He started with Wilcox’s safari shorts and went from there. “Everything just got bigger and bigger and bigger. And the shorts got smaller and smaller and smaller,” he says giggling.

Raised in a film-making family in Los Angeles, Gyllenhaal is reaping the benefits of time and opportunity. “[The longer you work as an actor], I think generally you care a little less about what people think,” he suggests, “and you’re allowed to present who you actually look to be or who you are.” He cites his recent appearance in a Broadway production of the musical Sunday in the Park with George as a case in point. “People seemed surprised that I knew how to sing, and I’ve sung my whole life!”

Of course, it is hard for any actor to succeed without support, and he says “working with people who want to work with you, who believe in you, is something I’ve only recently discovered. So often this business is all about chasing hype. A lot of it is that. And there’s just a point at which you say to yourself, ‘Am I going against the grain of who I am or am I moving in the tide I should be moving in?”

He hasn’t always listened to himself. Before the 2012 cop drama End of Watch brought him back to the kind of personal film-making that marked his early career, he veered into more obviously commercial territory, and lost himself. “I was unclear about what I wanted to say, and how I wanted to even just live my life, and where I wanted to be.”

Part of what got him back on track was the divorce of his film-maker parents, Stephen Gyllenhaal and Naomi Foner in 2009. They had raised him in what he acknowledges was a state of privilege, but at the same time had tried to give him and his actress sister, Maggie, a sense of perspective. His Jewish mother (his father is Christian), in particular, made sure they didn’t live inside a bubble, and for Jake’s barmitzvah, she took the family to feed children at a local homeless shelter.

“That was my mother being very conscious of giving us a perspective, and ultimately I think it has really influenced me,” he once told me. It was also, he half-jokingly suggested, his parents finding a compromise between their different faiths. “I think they wanted to share everything, all those ideas, so when it came around to having a barmitzvah, I think they split the opportunity and basically realised that in order to do that, ‘Well, let’s go feed the homeless’. Like that would be the most logical religious response to both Christianity and Judaism.”

He was 30 when they parted, and the end of their relationship seems to have been a part of their son’s creative rejuvenation. He moved to New York, and rediscovered the love for acting that he’d “displaced somewhere.”

With the divorce, he says, his whole family “started to shift and change. When people express their truth, I think it allows other people to do that. And when, after many years, they made that decision, and they went off and did other things in their own lives and established the lives that maybe they had always wanted, it allowed me, I think, to do the same thing.

Gyllenhall, 36, who is single and childless has obviously thought a great deal about families and how they work. “When parents are doing what they love and also loving their children, it allows their children, at whatever age, to express their truth.”

And truth is what Okja is about in as much as it’s a family film that doesn’t whitewash anything to spare the tender sensibilities of younger viewers.

Co-written by the Jewish journalist/author Jon Ronson (who based Gyllenhaal’s character on 70s TV personality Johnny Morris), it is about a young girl living in rural Korea with her grandfather and giant pet pig, Okja. One day, Wilcox arrives and has the beast shipped to America, where the company that genetically engineered Okja reveals its true colours.

Years ago, Gyllenhaal told me that he doesn’t believe anyone has a great childhood. “Even though we all pretend we want to go back and be children again, I don’t think we would really want to.” With Okja, he seems to have found the perfect expression of this.

“Bong would say that growing up is a brutal process. And that’s what this movie says: it’s as disturbing as it is extraordinary, but the only thing we have is the connections with the things that we love, and we should fight for those things as hard as we can.”

This message could hardly be more relevant in Trump-era America. Bong joked to Gyllenhaal about people fleeing to Canada after the businessman’s election, but the actor says he’s staying put.

“I think there is no better time than now to be in America and to work as hard as we can to express ourselves,” he says. “The last thing to do is run away. It’s a galvanising time. I have been more involved in what’s happening in my country, and more so in my community, than ever before. And I am actually inspired more so than ever.”

Okja has a limited cinema release on June 23 and will be available on Netflix from June 28