The musician smiles. "It's a very Mancunian trait - don't take yourself too seriously. It was more important for us to have a laugh than to have some great career strategy. If we enjoyed what we were doing, that was a great career strategy," he says.

Few bands have been as mythologised as New Order and their previous incarnation, Joy Division, while showing so little interest in mythologising themselves. Let others talk about how New Order rebounded from the suicide of Joy Division frontman Ian Curtis to become one of the greatest bands of the 1980s, or how records such as revolutionised the way rock groups assimilated electronic music.

Sumner's cheerful mood is down to the fact that New Order are, once again and most improbably, back from the dead. Barely a year ago, drummer Stephen Morris was saying: "There's no future for New Order." Yet here they are, amid an ad hoc world tour, a few hours before playing the closing concert of the Olympics in central London's Hyde Park, alongside Blur and The Specials.

New Order's latest resurrection has been marvellously unexpected, albeit coloured by a significant absence. When bassist Peter Hook quit in 2007, he declared the band over and Sumner formed a new group, Bad Lieutenant. But exactly a year ago came news that New Order would be reuniting, minus Hook but plus long-absent keyboardist Gillian Gilbert, for two benefit shows for music video producer Michael Shamberg, who was seriously ill with encephalitis. Those concerts went so well that they led to many more, including the recent Bestival music festival on the Isle of Wight.

So did they really believe last summer that it was all over? "Who knows what's around the corner?" Sumner says, deflecting the question. "I've had moments when I've thought, 'F*** them, I've had enough'. But once the anger dies down, the clarity pulls through. There comes this point where you've got this weight of history pushing you along, like a flywheel keeps an engine rolling round. You're not willing to give up the past 30 years of hard graft, blood, sweat and tears because someone else has had a tizzy fit, y'know?"

When Sumner is relating a favourite anecdote from the '80s he refers to the absent bassist as "Hooky", but in the current context it is always "someone else" or "the other one". New Order have had rifts before, which is why they have made only three albums in the past 20 years, but all the current members agree this split feels permanent, caused by something more than the usual friction between the emotional bass player and the drier, more reserved singer. When I talk to Phil Cunningham, guitarist in New Order and Bad Lieutenant for more than a decade, he says wearily: "It's not about Hooky and Bernard. It's Hooky and the rest of the world."

For the famously private Sumner, the last straw may have been Hook's public axe-grinding since his departure. His claim that replacement bassist Tom Chapman mimes to backing tracks and his sideswipes at Sumner in his new memoir are just the most recent examples. "He's not been very nice," Sumner says with acidic understatement. "To boil it down to its basic elements, if someone's horrible a lot of the time, you're not sad when they're gone, you're glad. He wasn't happy in the band - primarily with me but also, it seems, with almost everyone."

Of course, no outsider can ever know enough to take sides when old friends fall out. A clearly upset and self-questioning Hook told earlier this year: "I'm so confused with it all, and so obsessed with it all, I don't know even now what's going on." But without doubt his big mistake was believing the band couldn't continue without him, when New Order's very existence is testament to their extraordinary resilience. After Curtis killed himself in 1980, the surviving members of Joy Division swiftly reinvented themselves, with Gilbert (then, as now, Morris' partner) joining on keyboards and Sumner assuming frontman duties.

There have been other premature deaths over the years: producer Martin Hannett in 1991, manager Rob Gretton in 1999, and Tony Wilson, the quixotic boss of the Factory label in 2007. Has Sumner ever felt unlucky? "No," he says. "I feel lucky it's not me! Perhaps that's the reason we're better behaved because all those people having quite early deaths makes you aware of your own mortality."

Gilbert is equally matter-of-fact when discussing what kept her out of the band for so long. She took indefinite leave in 2001 to care for her and Morris' youngest daughter, Grace, who was diagnosed with the neurological disorder transverse myelitis. "New Order rehearsed at our studio, right next to where we lived, so I could still hear them," she says, taking Sumner's place at the pavement table while Morris attends to some last-minute technical emergency in Hyde Park and their eldest daughter Tilly loiters nearby.

In 2007, just as it was looking as if there would be no band to return to, Gilbert developed breast cancer. "I knew something was going on but when the break-up started I was going through chemotherapy and you just feel like nothing else matters. Now I've come out of that and stepped into this new line-up and we haven't got any of those issues any more."

If Joy Division's career had the stark precision of a novella, then New Order's reads like a baggy, tragicomic picaresque, studded with arrivals, departures, false endings and sudden twists. "In Joy Division, we did our learning in private, in dusty rehearsal rooms above pubs," says Sumner. "[New Order] had to learn in public. [a 1982 B-side] was really just an experiment to learn how to use a synthesiser. Instead of reading the instruction manual, let's write a song with it and, y'know, sell it to people. But when you try to play them live, you're a bit like, 'Oh God, where's the chorus? Oh right, there isn't a chorus'."

One collision of synth-tinkering and last-minute lyrical improvisation led to , their boldest and most beloved song. Sumner tells a story about being asked, in the '80s, to adapt the song to advertise Sunkist. Gretton, a man of staunch socialist principles, was in hospital and the money was great so they said yes. "The lyrics were something about when you're drinking in the sun, Sunkist is the one," says Sumner, laughing. "I was like, 'I can't sing that.' So Hooky wrote down '£100,000' on a piece of paper and put it on the mixing desk and said: 'Try singing it now.' So I sang it, and then Rob came out of hospital and was like, 'What have you stupid bastards been doing while I've been in hospital?' and put the kibosh on it straight away."

So there is no point expecting this strategy-averse band to know where they go from here. Perhaps there'll be a new album or perhaps this is just a cathartic and hard-earned lap of honour. At the moment, it doesn't appear to concern them either way.

For the first time in years, they are enjoying being in New Order.