On a cold but sunny day last month, a stream of nervous-looking players, male and female, ran on to a football pitch in the southern German town of Herzogenaurach. The match was an attempt to heal the wounds of a bitter family argument that has split the town for 60 years – and fuelled the fortunes of two of the world's most powerful sporting brands.

"The split between the Dassler brothers was to Herzogenaurach what the building of the Berlin Wall was for the German capital," says local journalist Rolf-Herbert Peters. Except that, whereas the Berlin Wall fell 20 years ago, the antagonism between Adidas and Puma is still obvious to any outsider visiting the town.

These two global brands were founded 60 years ago after successful shoemaker brothers Adi and Rudi Dassler fell out bitterly. They disbanded their 25-year-old company, the Gebrüder Dassler Schuhfabrik, which had made shoes for legendary athlete Jesse Owens among others, and formed rival manufacturers on opposite sides of the river Aurach, which runs through the centre of Herzogenaurach. And here the headquarters of these two giants remain today, barely a couple of miles apart.

What started the spat between the brothers is a point of contention. Town chronicles mention it only in passing as "internal family difficulties", but the most common explanation is that Rudi (apparently the better-looking one) had an affair with Adi's wife, Käthe, for which he was never forgiven.

But many other accusations fly, over who was the more enthusiastic Nazi (both joined the party in 1933), or who really invented the screw-in soccer boot studs that helped Germany's national team secure its World Cup final victory over Hungary on a soaking pitch in Berne in 1954. Many also point to a night in 1943 when Herzogenaurach was under allied bombardment. Adi and his wife apparently clambered into an air raid shelter to hear Rudi, who was already there with his wife and family, declare: "The Schweinhunde (pig dogs) are back." Adi insisted he had meant the RAF, but Rudi refused to believe him.

The enmity has divided the town ever since, determining which pubs its 23,000 citizens drank in, the butchers they frequented, who cut their gravestone and which football team they supported.

"There was a time when you'd have risked the wrath of colleagues and family if, as an employee of one company, you married the employee of the other," says Klaus-Peter Gäbelein of the local Heritage Association. "Even religion and politics were part of the heady mix. Puma was seen as Catholic and politically conservative, Adidas as Protestant and Social Democratic."

In business terms, it is Adi who has won. Adidas is by far the bigger company, employing 39,000 compared with Puma's 9,000. But it is the nature of the Adi- and Rudi-driven rivalry that has given both firms their fighting spirit, trying to outdo each other by securing endorsements with the world's top sportsmen and women.

"Listen, we have the fastest man in the world on contract," says Ulf Santjer from "underdogs" Puma, referring to Jamaican sprinter Usain Bolt. "We have the football world champions Italy, and Madonna wears our products – we're not doing badly."

Though the rivalry split the town, and meant that (officially at least) Adi and Rudi didn't speak to each other, even when Rudi lay on his deathbed in the 1970s and the priest called Adi to his bedside, the townspeople seem largely grateful for the rift. "Without the row we would not now be home to these two global players," says Gäbelein.

The Dassler brothers were building on Herzogenaurach's manufacturing traditions – which started in the middle-ages when it produced textiles and, later, shoes and carpet-slippers – when they began their joint enterprise on returning from the first world war. Their father Christoph, a shoemaker, passed on the tips of the trade, and so it was that Adi gathered the tools and equipment left by retreating first world war soldiers and "shaped their first shoe, a cross between a carpet slipper and a running shoe", while Rudi "took on distribution and general management", according to Rolf- Herbert Peters, author of The Puma Story. Their mother's laundry room became the company's headquarters. "The cramped laundry . . . housed the nucleus of a future global company which would revolutionise the sport and fashion worlds," says Peters.

Neither company is now controlled by descendants of their founders – the brothers are long since dead, and Puma is majority-owned by the French luxury goods maker PPR, while Adidas is owned by lots of small shareholders. Nevertheless the companies still keep their headquarters in Herzogenaurach. Manufacturing may have long since moved to distant corners of the world, but the factory workers have been replaced by young, sporty employees from many nations who concentrate on design, marketing and brand awareness and communicate in English. This town of cobbled streets and half-timbered houses still matters to both companies. It's almost as if moving away to nearby Nuremberg or Erlangen would be admitting defeat.

"It's always good to have a heritage – it's your history and your experience combined," says Kirsten Keck, Adidas's spokeswoman, showing me around its World of Sport. On display are the friction-stretched long-jump shoes worn by Owens at the 1936 Olympics, gymnast Nadia Comaneci's white floppy Adidas gym shoes in which she was awarded a 10 in Montreal in 1976, and Muhammad Ali's customised high-ankled boots, worn when he fought George Foreman in 1974. On the wall is a photograph of Adi and Rudi, pre-fall- out, playing ice hockey on the frozen Aurach river which would become the dividing line between their companies.

Both businesses are expanding their presence in the town, building plush "brand centres", catwalk-style complexes and showrooms of their most successful products. In a time of recession, the cranes that dominate the skyline as workmen finish the Adidas complex, Laces, and Puma's new establishment, Plaza, are a welcome sight, and the message is clear: "We're here to stay" – and, of course: "We will not be outdone".

The wonderfully named mayor, German Hacker, tells me that, even when it came to the recent football match, it would not have done any good for the town's morale to have one "firm" beat the other. "This match was diplomatic down to the last blade of grass." Even the ball was co-branded. (It is somewhat disappointing to find out it was a "management versus workers" game, which the bosses won 7-5.)

"For anyone who grew up here this was an extraordinary affair," says Hacker. "It would have been impossible 30 years ago. But the mood in the town has changed. The families no longer run the companies and we have people from 85 different nations living here now, everyone from graphic designers to engineers, and the companies are run by two young and dynamic heads."

What the town is crying out for, but will probably never get, is a shoe museum. "The companies would never be able to decide on a common story," smiles Gäbelein. Instead, visitors are left to their own devices. In the stillness of the town's cemetery, few of those tending the graves are even aware that both brothers are buried here. But after some time I find them, Adi's granite gravestone nestled in a bed of purple pansies and, at the opposite end of the cemetery, Rudi's final resting place, marked by an Angel carved in stone next to a fir tree.

And yet, before they died, the brothers may, in fact, have been reconciled, according to several people I meet (some of whom only dare speak about it in whispered tones).

"In 1974, just six months before Rudi's death, they got their drivers to take them to a secret meeting in Nuremberg for half a day," says Helmut Fischer, the peace match referee and Puma's in-house historian, referring to conversations he has had with those closest to the inner workings of the Dasslers' empires, including their chauffeurs and housekeepers.

"But they could never tell their wives, and certainly not their workers, because it would have been bad for business."