Ten years ago, the angry young man of punk and legendary frontman of the Clash, Joe Strummer, died suddenly from an undiagnosed heart defect. Instead of celebrating what would have been his 60th birthday next month, his two daughters Jazz and Lola, will be marking his life and legacy along with 5,000 others in a field in Somerset at a one-off music festival they have helped to create, called Strummer of Love. It is a fitting tribute for a man famed for his love of a good music festival, and who brought up his daughters immersed in the same world.

From the moment they were born, both Jazz and Lola accompanied their father on his annual pilgrimage to Glastonbury. "We grew up going to all sorts of festivals," says Jazz. "Every year at Glastonbury, Dad would create this kind of impromptu camp where people would just gather.

"He was always setting up camps – even with us as kids, it was always I'll make you a camp out of sofa cushions and stuff. He loved that whole idea of creating a vibe and an atmosphere. At Glastonbury he would string up his flags, get loads of hay bales, have a big campfire – there'd be 24-hour music and he'd be DJ-ing. We'd all go as a family, and our friends would be there. It just became this hub of fun and people. He named it Strummer- ville and we were left to do exactly what we wanted. Run wild, run amok."

At the time of Joe's death, a few days before Christmas in 2002, Jazz was 18 and Lola 16. Their father collapsed at his home in Somerset after taking his dogs for a walk.

"I remember I was in Oxford Circus trying to do some Christmas shopping," says Lola. "You always kind of know, I think. I was sort of in distress that day and couldn't figure out why. I got a phone call saying come home. So I got on the tube and I remember sitting there, weighing up the options. I knew it wasn't my mum because I had spoken to her earlier, and so I thought it must be either Jazz or my dad. By the time I got off the tube I rang home and said, 'Dad's dead isn't he?'"

"It was such a shock. It wasn't like he'd been ill. The day before, we'd all had such a great day with him. He had been away on tour and we hadn't seen him for a couple of months. So we all met up – our mum, our grandparents, his second wife Lucinda and her daughter Eliza, and we'd all gone out for a meal and then sat in the Groucho [club] drinking champagne. It was a really, really lovely day."

The girls reacted in very different ways to Joe's death. For Jazz, it took a while to sink in properly, and a few years afterwards she experienced panic attacks. "I think I had a bit of a delayed reaction," she says. "I went to see someone and we talked about it a lot. Now I feel quite resolved."

Lola dealt with it more immediately. "After it happened, I completely let myself deal with it," she says. "I was really miserable and it was a very tough time. Jazz was living away from home and I felt there was no one really around. It changed my life completely. But you do get over it. Death is just a part of life and you have to accept that."

Both girls say that helping to organise Strummer of Love has brought many feelings flooding back. Everyone in the lineup has been chosen because of a special connection with Joe Strummer in some way. So among the performers are Mick Jones, the former Clash guitarist, the Pogues, with whom Strummer also played, Alabama 3, with whom Strummer's stepdaughter Eliza now sings, and Billy Bragg, his long-time friend and political ally.

"It's kind of strange because it has been 10 years," says Lola, "but I feel like a lot of stuff is resurfacing – feelings, almost like grief. It's weird. And it's stronger now than it was just a few years after he died."

Jazz feels the same. In June this year she gave birth to her first child, Boudicca – who would have been Joe's first grandchild. "Having a baby makes you rethink a lot of stuff, and I've been thinking about Dad a lot recently," she says. "He would have been obsessed with her. He was fantastic with kids. He loved them, he really did."

Still, both sisters are acutely aware that their father's heart defect could have cut his life short at any moment, so they are grateful for the time they did have with him. Plus, there is also his musical legacy.

"I'm just grateful for his music because we still have his voice," says Lola. "We are lucky to have that."

Joe also did a series for the BBC World Service called the London Calling broadcasts, to which both girls listen regularly. "I remember he took us to Bush House and let us sit through the recording," says Jazz. "They are great because it's him talking and picking his favourite tracks, which brings a lot of his personality out – which is nice because that's the kind of thing you forget. I like to listen to them when I'm working in the studio, and sometimes when we have parties we put them on. It's comforting."

Neither of the sisters got to see the Clash as the band started disintegrating in the early 1980s, just before Jazz was born in 1983.

At first the family lived in Ladbroke Grove, London. Joe, a diplomat's son, had been sent to boarding school aged nine and tried to give his daughters an upbringing different to his own.

"As kids we were really encouraged to be free," says Jazz. "Dad came from a strong authoritarian background. His father was very academic, and he went to public school, which he found really tough. He hated having that put upon him as a kid, so he tried to encourage us to be as freewheeling as we liked.

"When were growing up there were no rules – we were left to run wild. We were nicknamed the pit-bull kids because we were so mad. At home we were allowed to scribble on the walls because he considered it creative. We'd ransack the place.

"I think our mum might have had a bit of a different view but she kind of went with it."

The upshot of Joe's liberal attitude, however, was that Jazz found herself expelled from nursery school, when she was barely out of nappies. "I had slight behavioural problems," she says. "I'd just throw my clothes off and run around. I was disruptive and unruly. It's really embarrassing, actually. In the end, they couldn't get any school to take me so we moved down to Hampshire where my mum found a nice little private school that would let me in."

Being Joe Strummer's offspring wasn't always plain sailing. When Jazz and Lola were around eight and six, their parents divorced. "We both dealt with it differently," says Lola. "I think I was quite an oblivious child – I didn't really have a clue – but I do remember Jazz cried, so I cried."

"I don't think I dealt with it very well at all," says Jazz. "It was pretty horrific. But by that point Dad was quite vacant – he wasn't really around much anyway. I think he was very unhappy and frustrated creatively."

The years when Joe's career was floundering had a big impact on his family. "There was quite a dark period, when we were a bit older and he couldn't get work, and he was struggling," says Lola. "It was hard for him to move on musically and creatively. He'd be at gigs and people would just be screaming for Clash tunes – but if you listen to his music, his tastes had completely changed. He'd mellowed and softened. When he died, though, his career was starting to take off again. I remember thinking it was such a shame. But in a way it was better that he went out on a high."

Despite the divorce, their parents maintained a good relationship and the two sisters have fond memories of the adult relationships they formed with their dad. Every summer they would go to Spain, to Joe's home in San José, Andalucía, a part of the world he had fallen in love with.

"When we got to a certain age, he'd just take us out drinking with him," says Jazz. "We'd go pub crawling round all the little Spanish bars; he'd invite all our friends, too, and we'd just stay up all night. He was so generous and welcoming. He was a real Pied Piper character."

There was talk about the Clash reforming before he died. "But there had been talk for years and years about them reforming," says Jazz. "They had been offered stupid amounts of money to do it, but they were very good at keeping the moral high ground and saying no.

"But I think if Dad hadn't died, it would have happened. It felt like it was in the air," she adds.

It is clear that Joe Strummer's creativity has rubbed off on his two daughters. In 2004, they started organising club nights in East London, at Bethnal Green Working Men's Club, kick-starting the trend for rock'n'roll tea dances. Jazz is also founder of the successful Shoreditch arm of the Women's Institute and last year her book on sewing and baking, Queen of Crafts, was published.

Lola sings in a band called Dark Moon, which is playing at Strummer of Love and Bestival, and designs clothes for her label She Vamps. The two sisters share a studio in east London, near where they live.

"I started the WI because I had got really impassioned about women's issues and women's rights," says Jazz. "I think because the Clash were so political and Dad had such strong opinions about equality and stuff, it affects you. But, equally, he was a great inventor of things – he was always making things happen. So I think more important for me was his passion. He'd have an idea and then do whatever he could to make it happen. I think that's what really rubbed off for me."

At Strummer of Love, Jazz will be looking after her eight-week-old baby at the same time as organising a big DIY tent called the Handmade Hangout, where all manner of craft classes will be taking place. Lola, meanwhile, will be singing from the pop-up stage, and their mother has organised the healing field.

"It's funny looking back, but I always had a feeling he must have known what was going to happen to him," says Lola. "Our stepmum found these lyrics he'd scrawled on a piece of paper a few months before he died, saying, 'I was just somebody, who loved a body then left a body' or something like that. I took them and turned them into a song because I felt it needed to be written. But I do believe he knew."

Either way, as his family roll out the hay bales and gather together round the Strummer of Love campfire, Joe's spirit will still be with them. "I just know he'd love that we are doing this celebration for him," says Jazz. "It would be totally right up his street and such a good expression of who he was. He would have loved it."

www.strummeroflove.com