Robert Redford was the pretty one, James Dean the tragic one, Steve McQueen the rugged one and Marlon Brando was the wild one. But Paul Newman was the perfect one; beautiful but also masculine, with such a pretty surface that had obvious depths beneath. He was as good playing the tough guy (Hud, Cool Hand Luke, Cat on a Hot Tin Roof) as he was at emotional vulnerability (The Verdict, Road to Perdition), and no one had more instant, can’t-look-away screen charisma (Butch Cassidy and the Sundance Kid, The Sting, The Color of Money).

On top of all that, his 50-year marriage to Joanne Woodward was famously happy and his philanthropy was so extraordinary that when he died, the Economist wrote that he was “the most generous individual, relative to his income, in the 20th-century history of the United States”. As I said, perfect.

So when news emerged that an unpublished memoir, once thought lost, had been discovered by Newman’s family and would be “revealing and surprising”, fans braced themselves for the worst. Were we to be subjected to tales of Paul Newman, abuser of small animals? Paul Newman the devil-worshipper? Must every hero fall?

But Paul Newman: The Extraordinary Life of an Ordinary Man, which was published last week, only burnishes his reputation. Not because he bangs on about his glorious deeds – in fact, he barely mentions them at all – but because this once very private man reveals so much of his fascinating, never disappointing self. He was constantly questioning himself, riddled with guilt about the past, deeply in love with his wife and trying to learn from his mistakes. Human, in other words, but better.

One of my favourite passages, typical in its self-awareness, is when he describes how badly he handled his children from his first marriage, Scott, Susan and Stephanie, when he left their mother, Jackie Witte, for Woodward: “What I did just didn’t have any class. I didn’t take them aside and give them comfort by explanation, certainly not in a way that they would understand. Not because I wouldn’t do it, but because I didn’t really get it.”

Clea Newman Soderlund is the youngest of Newman and Woodward’s three daughters and, like her older sisters Nell and Lissy, she has her father’s bright blue eyes – a gift of an inheritance. “One of the biggest surprises for me when we found the transcripts was how hard he was on himself. That’s difficult for any child to read and it’s not the way I saw him,” Soderlund tells me from her home in Connecticut.

She remembers clearly the long conversations her father had in the mid-1980s with his close friend Stewart Stern, the screenwriter for Rebel Without a Cause, which they intended to turn into a memoir. Newman also asked Stern to interview those who knew him best, including both his wives and many people he worked with, such as Elia Kazan, Sidney Lumet, Tom Cruise and George Roy Hill, so the book would give the whole, true picture of him.

“That process was really like therapy for him, and I got the benefit of it in the years afterwards, because, while nobody’s perfect, he really was wonderful to be around. Just so present, showing up for everything. He knew this work was important, and he used the lessons he learned from it,” Soderlund says. The transcripts of the interviews, however, were misplaced and when Newman died in 2008, followed by Stern in 2015, the project seemed lost for ever – until the pages were found carefully filed in a cabinet in Newman and Woodward’s house. Newman had stipulated in his will that his children had permission to publish a biography of him, so putting out the lost memoir, they felt, was following his wishes. “He wanted to get everything out there,” Soderlund says.

To his children, Newman was “like Superman”. But in his book he contrasts the godlike image people had with who he was inside, “unexplored, uncomfortable and unknown”. He describes his problems with alcohol and the pain he experienced from antisemitism, because his father was Jewish. When his acting career was taking off, it was suggested that he make his name sound less Jewish, as many other actors have done, from Kirk Douglas (Issur Danielovitch) to Tony Curtis (Bernie Schwartz). But Newman refused: “It seemed more of a challenge to me to keep my real name, to insist upon it as a badge,” he writes. “That was typical of Dad, to go for the harder option and then work to overcome it,” Soderlund says.

After serving in the US Navy in the second world war, Newman made his way to the famed Actors Studio in New York, where he was taught alongside James Dean, Ben Gazzara and Julie Harris. “I was IN their world but definitely not a part of it,” he writes. Dean teased him when Newman’s first film, The Silver Chalice, flopped and Dean’s debut, East of Eden, was a triumph. After Dean died, Newman was given Dean’s part in Somebody Up There Likes Me, which became his big break. “I know there are some people who attribute my career breakthroughs to Jimmy’s death. Yes, there were elements of luck,” he writes, although he adds that “half of him” believes he’d have made it anyway. But only half.

Newman married Witte in 1949 but in 1953 he met Woodward. The attraction was instant, and he credits Woodward with giving him onscreen sex appeal. “Joanne gave birth to a sexual creature. I’m simply a creature of her invention,” he writes, and describes how they left “a trail of lust” everywhere they went. Eventually he left Witte and married Woodward in 1958. In one of the most memorable parts of the book, Newman describes coming home to find Woodward standing in a freshly painted room with a double bed. “‘I call it the Fuck Hut,’ she said proudly,” Newman writes with palpable delight.

Did their children know about the Fuck Hut? “Ummm, no,” Soderlund says with a pause and then a smile. “But when I read it, I was like, ‘Go Mom!’” Their passion for one another was never hidden, she says. “To get into their bedroom, they had one ordinary wooden door and then also a really thick door, and both locked on the inside,” she says and laughs. “It was a very sexy relationship. It wasn’t a fairytale, but he was really, really in love with Mom.”

Not long before Newman died, when he was having chemotherapy for lung cancer, Soderlund went out for dinner with her parents. Afterwards, she offered to walk them to their car in the rain. “But my dad waved his hand, like, ‘I got it.’ I watched them walk to the car, him with his arm around Mom’s back and holding the umbrella above her head, then opening the car for her. And yet, he was so sick. It was such a juxtaposition between the horrors of getting old but also the wonders of being old and having a relationship like that,” she says. He was 83 when he died.

Woodward, 92, still lives in their house in Connecticut, but has had Alzheimer’s for several years. “Losing Dad just floored her. The world stopped for her then. I remember going to an event with her in New York soon after my father passed away, and people crowded around her, caring about her, but she became fearful. I was there, my husband was there, but it wasn’t the same, because he wasn’t there,” Soderlund says.

Newman had, by any measure, a good life, but one that also bore a deep scar. In 1978, his son Scott, who had struggled with drugs for years, died from an overdose aged 28. “I want to embrace the responsibility for that. What would it have taken to avert that? I don’t think I could have gone into films and been a movie star. I couldn’t have drunk. I couldn’t have been a risk taker. I think that’s enough for me to say right now,” he writes.

Newman is far from the only movie star to have lost a son to an overdose or suicide. From his era alone, there were also Clint Eastwood, Gregory Peck, Louis Jourdan and Kirk Douglas. “Dad was right that growing up with parents who are artists and are really well known isn’t the easiest of things in the world. There are some great benefits, and also some very not great ones. Also LA in the 60s and 70s was kind of a reckless place. There were all these drugs that were readily available, and the parents were from a generation that had no understanding of them at all,” Soderlund says.

Did Newman ever talk about Scott? “Not very much. I think it was too raw for him until the day he died.”

Newman channelled his grief into founding the Scott Newman Center, dedicated to preventing drug abuse. To list all his philanthropic endeavours – not least the famous spaghetti sauces and salad dressings – would take a whole newspaper. Soderlund is still the ambassador for SeriousFun, Newman’s network of summer camps for seriously ill kids. Because of the efficacy of his endeavours, plus the amount of money he donated, he changed the way many people perceived celebrity philanthropy, although he plays it down in his memoir: “The easiest thing I can do, frankly, is to give away money,” he writes. One of Newman’s only regrets when he realised his cancer was terminal was that he still wanted to do so much to improve the world. “He just wasn’t ready to go,” Soderlund says, crying now.

Newman embarked on philanthropy for the same reason he embarked on this memoir project. To everyone else, he seemed perfect, but Newman felt he knew the truth. “I think what’s really amazing about him is that he kept growing and working to be a better person as life went on,” says Soderlund. “He used to say, ‘We Newmans are late bloomers’, and he kept trying to be the best him. Isn’t that what we’re all striving for?”

• Paul Newman: The Extraordinary Life of an Ordinary Man is published by Century.