"I inherited my sister's kimonos and fans, which became my favourite clothes," he says today. "And though I had never known her, I inherited her gestures as well."
This may sound like pop-psychology with hindsight, a too-neat rationalisation of the vamps and divas, the wild-eyed virgins and bloodstained femmes fatales, whom Kemp has spent the past 30 years incarnating on stage. But personally I buy it, as I buy the equally plausible contention that the male characters he plays - from goose-stepping generals to sailor boys - derive their force from his naval-officer father's premature death at sea. Family myths are the most potent of all.
But it has to be said - and he would be the last to deny it - that Kemp is such a self-mythologiser that not even he can now sift fact from fiction. Did he go to prep school in a kimono, or did he simply turn his navy-blue cape inside out to display himself in its red satin lining? No one knows, but either way it incurred displeasure. At boarding school he got up on pointe in his football boots and gave nightly cabarets in the dorm. "I was so different from everyone else, I had to enchant to save my skin. It was there that I learnt the art of hypnosis, which is an essential part of the artist's technique."
As he tells it now, while rehearsing its newest manifestation for a show at the Hackney Empire, the myth develops with sweet inevitability. He trained with Queenie Roma and her Sunshine School of Dancing, and saw his first real ballet in company with the young and undiscovered David Hockney. He joined the Royal Air Force as a medic ("I always felt I belonged to the healing profession") and did a stint with Ballet Rambert before getting kicked out for being temperamentally and physically unsuited. ("Imagine if I'd taken notice! What the world would have been deprived of!") While sharing a flat with the young and undiscovered Steven Berkoff, he worked in a Shaftesbury Avenue steakhouse. "But I didn't last long there, because my speciality was free meals for actors." He got kicked out of a teaching job at the Guildhall for "fraternising" with the students ("Well, I couldn't just start and stop by bells").
He confesses that he's forgotten which of his stories about meeting David Bowie is the truth, but it's documented fact that he put Bowie on the map as Ziggy Stardust. "David had originally wanted to be a monk, but I saved him."
By this time, in the Seventies, Kemp himself was on the map, and he's been there ever since, growing steadily more luminous in the light of world acclaim. This acclaim is not universal - purist ballet critics loathe him - but theatres in America compete to get him, the Japanese roll out the red carpet (his new show is backed with Japanese money), and in his adoptive city of Rome he's regarded as home-grown royalty. His retinue of Kemp-followers - middle-aged ladies, and starry-eyed kids of both sexes just out of school - is constantly renewing itself, and sometimes throws up stars of its own. He remembers one eager little girl running round making coffee when they were doing Salome at the Round House: Kate Bush, before she was famous.
What draws them all is the exotic blend of danger and innocence, of palpitating femininity and sturdy athleticism which were Kemp's trademarks from the outset. Not quite dance, not quite mime, occasionally spoken, often crazily sung, his brand of theatre digs deep into the subconscious. It's endearingly low-tech, its images are presented with colourful abandon, and it's set to music by a composer - the Peruvian Carlos Miranda - whose scores perfectly echo Kemp's eclectic splendour. In each production Kemp himself is the focus - occasionally too much so, as in the Kabuki-inspired Onnagata - but when a show works, hypnotic is the word. Puck was never more ethereal than as Kemp plays him, sliding serenely down a wire across a garden of Arcadian delights. Erich von Stroheim would have recognised his psychopathic double, as incarnated by Kemp in The Big Parade. And Jean Genet would have been mind-blown by Kemp's ecstatic parody, surrounded by an attendant troupe of boy-angels in Flowers, of the femininity he both hated and adored.
It was Flowers - staged at the Bush in 1974 - which defined its creator's style, and which for two decades has remained his flagship production. Drawing inspiration from the nightmare visions of Fernando Arrabal - which made Artaud's "theatre of cruelty" look like a vicarage tea-party - Kemp went into the mean streets of Edinburgh and recruited the most charismatic young vagrants he could find. The show progressed from basement venues to main houses, and on to a protracted world tour. In Sydney, Kemp was met by the massed ranks of the Army of Light, praying for their city to be kept free of moral taint. In Barcelona, just after the fall of Franco, the show was hailed as a symbol of Spain's liberation.
Cinderella, the last show he brought to Britain, was a worthy successor in the tradition; alarmed by the fact that the heroine got raped by her father, Sadler's Wells billed it as unsuitable for children under 16. "What did they expect?" asks Kemp with faux-naif wonder. "Everyone knows that's the story, even if you don't see it in the Walt Disney version, or the one by the Royal Ballet. I tried to remind the management that a bit of scandal never hurts the box-office, and that in any case people know what to expect of me. They know it's going to be Kempian, which always means a bit of incest. I do have this big responsibility, not only to enchant people, but also to surprise them."
There was a sulphurous whiff about the company in their youthful prime, well captured in a book by the Berlin photographer Anno Wilms. This opens with a preface by Derek Jarman (who used Kemp's home-grown stars in film after film): "What David Hockney did for the art world, Lindsay did for the theatre." There follow extraordinary pictures of founder-member Neil Kaplan, rangy Australian clown Michael Matou, and the Brazilian Atilio Lopez, a spot-on Douglas Fairbanks in The Big Parade. All have died of Aids.
"They were super-talented," muses Kemp as he watches his present company - bursting with coltish energy - limber up on mats and trapezes. "The company lost its backbone. My new people are starting out with skills the original lot took years to acquire. The question is, will they have the same genius? But we auditioned hundreds to get them: the new show is an all-male-voice musical, and time and again we found that someone who looked right, and maybe acted and juggled brilliantly, didn't have the requisite falsetto. It's just my luck that the only real woman should be playing a part which is deaf and dumb." That's Nuria Moreno, who took a three-month job as a casual helper 16 years ago, and is now a star. In this company, Kemp says, everybody does everything. "In A Midsummer Night's Dream, even the electricians become fairies."
The new show is a circus tragi-comedy entitled Variete. Dedicated to Kemp's hero, the film director Federico Fellini, it's inspired by Buchner's Wozzeck, Charlie Chaplin's The Circus and Tod Browning's outrageously un-PC film Freaks. Even on day two of the rehearsals, the expressionism of its opening scene comes across strongly: just right for the crumbling Hackney Empire, and an interesting prospect for Poole and Barnstaple, where it will later go on tour. This is the first time Kemp has had the luxury of Equity pay and conditions, and he's revelling in it.
Will he go soft as a result? "Not a chance." But there was a time - not so many years ago - when he had gone soft, incapacitated by drink off- stage, sometimes even on. Is he now reformed? "Totally!" And he does a neatly executed double spinning kick to prove it. Until last week, he points out, he was touring in one of his own ballets with the Teatro Nuovo di Torino: sharing the stage with people one third his age, he had to keep superfit. He now does a 90-minute class with his company every morning.
For several years, Kemp and his literary manager, David Haughton - one of the original boy-angels - played an active part in Italian politics, doing television ads for a variety of crusades. That stopped when their party joined up with Berlusconi, but they still campaign unpaid if the cause is right. Haughton, who recently got married ("To a lady!" gasps Kemp, in tones of mock-outrage), now directs theatre and plays villains in Hollywood epics. Carlos Miranda, who composed the music for the Barcelona Olympics, is planning his South Bank coming-out as a composer tout court.
Kemp, meanwhile, has just bought an Umbrian monastery to provide his company with a base, but he is also embarking on a career as an opera director. His first production, of Rossini's Barber of Seville, was a popular success, but it got a sniffy response from the Italian critics. He'd stuck too faithfully to the text, they complained, and it wasn't Kempian enough.
These boys are still full of surprises.
'Variete' is at the Hackney Empire, London from 24 Sept to 5 Oct. Booking: 0181-985 2424