So what is it about this ballet? Given that only a fraction of any house can be first-timers, that leaves an awful lot of punters with a serious annual craving for animated dolls, fighting rats and dancing snowflakes, and not all of these people bring children. The simple answer, of course, is the music. Tchaikovsky's score offers two solid hours of the most scintillating tunes and orchestrations; music of 24-carat quality that remains untarnished by endless repetitions in the theatre, and even by the mucky thumbprints of the advertising industry. The image of Frank Muir singing "Everyone's a fruit-and-nut cake" has not left a permanent mark on that felicitous solo for piccolo. Nor have clod-hopping parodies by TV comics managed to dull the inspired pairing of celeste and bass clarinet in the Dance of the Sugarplum Fairy.
Something about Nutcracker as dance has roused the talents of almost every top-ranking choreographer: Ashton in 1952, then Balanchine and Cranko, Nureyev in 1967 and 1968 (different versions for different companies), Baryshnikov, Petit and Schaufuss, our own Sir Peter Wright (who's also notched up two, both currently on the go), and droll post-modernists Matthew Bourne with Adventures in Motion Pictures, and the American Mark Morris.
Again, they would say, have said, it was the music that fired them. And the desire to serve it better. Tchaikovsky's score, written closely to a scenario suggested to him by the director of the Russian Imperial Ballet, is in fact all that remains of the original Nutcracker of 1892. Details of the steps were not recorded. Textbooks will tell you that what you see today in the final Grand pas de deux are close to those devised by Lev Ivanov, Imperial Choreographer, but as to even that tiny fraction of the whole, ENB's director Derek Deane says, with an airy shrug, "Who really knows?"
"We say, 'choreography by so-and-so, after Ivanov,' out of politeness really. As there was no form of notation at the time, nuances may have survived, but with the Russian classics we're entirely reliant on old dancers' memories of what another dancer taught them of what another taught them - and when you're talking about something as detailed as steps, that's a pretty weak link with a 100-year-old past."
Two years ago, many British ballet- lovers believed they were going to get as near as dammit to the real St Petersburg Nutcracker when the Kirov company flew in for Christmas at the Coliseum. Sleigh bells, winter palaces, proper snow - the Russians know about these things. The Kirov is also the direct, post-Revolution, post-Lenin descendant of the Imperial Ballet, and thus claims a more or less unbroken link with its 19th-century repertoire. The result was desperately disappointing ("No, hideous, horrendous!" says Deane), with ugly costumes, shaky sets, acting that lurched between simpering and ham, and a midnight battle of the toys that had all the dramatic force of a tiff between Sooty and Sweep.
The eye-opener of the Kirov fiasco was how, in Russia, a full evening's story-ballet appears to be first and foremost an opportunity for bravura dancing. Concern for character development, motivation and continuity barely seemed to register. For these are values absorbed from mainstream theatre and opera, and a relatively recent development in classical dance. The great Russian companies, immured in their own sense of heritage and cut off from Western influence by Communism, have missed out on all that. The Kirov Nutcracker quite probably did, as it promised, adhere to Ivanov's "original libretto". But it so lacked the narrative coherence we now expect when we go to the theatre, it came across as empty and crass, despite some dazzling dancing.
Nutcracker has avoided falling into the "authentic" trap in the West principally because it didn't appear outside Russia till 1934, when Ninette de Valois staged Act II for her young Sadler's Wells troupe. It didn't appear in its full version until the 1950s. Up to then, Tchaikovsky's score was known to us only as the truncated orchestral suite he devised in a shrewd act of marketing from the "national" dances of Act II. Even now, some people think the Nutcracker Suite is all there is.
Walt Disney used this latter to magical effect in his 1940 animation film Fantasia, a work of imaginative visualisation that reached far beyond ballet's regular constituency. The sultry Arabian dance becomes a play on the flickering, slinky movements of exotic fish; the Chinese dance a ballet for button-top mushrooms. The famous Waltz of the Flowers is a ravishing choreography for water lilies. No stage realisation of this section of the music has ever capped Disney's, though Matthew Bourne's for AMP came close. His Kingdom of Sweets had Doris Day marshmallows, gob-stopper skinheads, and a hashish-smoking Knickerbocker Glory. But his gorgeous treatment of the snowflake dance was a knowing wink at another ballet altogether - Ashton's suave skating number, Les Patineurs.
Choreographers have struggled with a scenario that is awkwardly unbalanced: a first half which is all plot and no dancing, and a second which is all dancing and no plot. It hadn't been easy for Tchaikovsky either. His choreographer had dumped on him a diluted, sugared-up version of a story by the German writer ETA Hoffmann, a dark and grotesque piece of gothickry originally called Nutcracker and Mouse King. What emerges in Tchaikovsky's hands is a warm evocation of a Christmas party, followed by a brilliant, almost apocalyptic vision of magical transformation, the real world into dream world, the Nutcracker doll (such a very odd present for an adolescent girl) into a strapping young Prince.
It was Nureyev, in the late 1960s, who first brought Freud into the picture, by making the mysterious toy-dispensing godfather Dr Drosselmeyer turn into the yearned-for Nutcracker prince. Since then, no choreographer worth his salt has presented the plot without some reference to Clara's sexual awakening, whether by having her openly swoon over her cartoon dream hunk (as Matthew Bourne did), or more subtly, presenting her with a Christmas gift of point shoes, underlining her desire to move on from childhood while preparing us for the stretches of stage time that are about dancing for dance's sake.
ENB's new Nutcracker, reworked by Derek Deane last year, makes a deliberate bid for an adult audience. It presents the Christmas party in a chic, sterile Belgravia-style mansion - a place where childhood is a thing you buy at Hamleys. Drosselmeyer is a louche, Gucci-clad guest who takes the kind of interest in Clara that would put most mothers on red alert. Although in Act II he ostensibly offers the girl no more than a vista of dancing pick'n'mix, his demeanour speaks otherwise: hands straying over her shoulders, mouth hovering near her throat, overseeing her duets with the Prince like a dirty old voyeur. On the production's first outing, most critics remained to be convinced. This year, Deane has courageously abandoned his nine-year retirement from the stage to dance Drosselmeyer himself, and show us what he was getting at.
The alternative at BRB presents a more conventional balance of tradition and innovation, presenting Clara with a happy home and dreams that are innocent and idealised. But its overall scheme has a seriousness and coherence that makes perfect sense of its peculiar parts. The dancing is spread evenly throughout, and the transformation is the most riveting you'll see. The Christmas tree grows gigantic, but so does the fireplace, out of whose fiery innards burst an army of rats. Artistic director David Bintley confesses that his own son has had nightmares about those.
"In the end," Bintley says, "the joy of Nutcracker and the reason it endures is that it can be made to speak to all ages. The plot must touch something in our psyches about being small, about adults having power in our lives. You find it in Alice, in The Borrowers, in Gulliver ... It's powerful imagery that resonates with everyone. I have to watch an awful lot of Nutcrackers in my present job, but for the first few of every season I still have a moist eye."
Yet Bintley, our most successful creator of narrative ballets, inherited BRB's current show from his predecessor, Sir Peter Wright. He has never set his hand to the Nutcracker, though he did produce a hugely enjoyable setting of Duke Ellington's jazz version of the Suite, a jewel in BRB's crown, shown on television next Friday. Has he never been tempted to pay homage to Tchaikovsky? Bintley has thought about this a great deal. "One of my fondest ambitions," he says with care, "is to write something, anything, that could replace it."
ENB: Coliseum, WC2 (0171 632 8300), to 9 Jan. BRB: Lyceum, WC2 (0870 606 3441). 'Nutcracker Sweeties': BBC2, 5.30pm on Christmas Day.