'Mild protests against the music," wrote Stravinsky, "could be heard from the beginning." The composer was remembering the night of 29 May 1913 at the Théâtre des Champs-Elysées in Paris. The event was the premiere of a new ballet called The Rite of Spring – and, if you believe all the stories about what happened that celebrated evening, not least the one about the riot that ensued, it's as if the 20th century only really got going when the audience in that gilded art-nouveau auditorium started kicking off.

If you know how Stravinsky's music begins, you may not be too surprised by the audience's reaction to The Rite, which was choreographed by the young dancer Vaslav Nijinsky and performed by Serge Diaghilev's Ballets Russes. After the strangest, highest and most terrifyingly exposed bassoon solo ever to open an orchestral work, the music becomes a sinewy braid of teeming, complex woodwind lines. "Then," Stravinsky told his biographer, "when the curtain opened on the group of knock-kneed and long-braided Lolitas jumping up and down, the storm broke."

That was Nijinsky's choreography for the Dance of the Adolescents section, the music's first and still-shocking moment of crunching dissonance and skewed rhythm. Stravinsky said that at this point, "Cries of 'Ta gueule' [shut up] came from behind me. I left the hall in a rage. I have never again been that angry." Stravinsky spent the rest of the performance in the wings, holding on to Nijinksy's tails as the choreographer shouted out cues to his dancers over the din.

What really happened on that night of nights? Was this a genuine riot, as it is so often described – a shocked response to Stravinsky's simultaneously primitivist and modernist depiction of an ancient Russian ritual devoted to the seasons? Or was it simply a publicity stunt, a wilfully orchestrated succès de scandale that has, in the years and the retelling, grown into a great musical myth? And was The Rite really such a revolution in music, a gigantic leap of faith into a terra incognita that would inspire every subsequent composer?

There is still no more influential piece of music in the 20th century. The Rite is the work that invariably tops polls of the biggest and baddest of the last 100 years. From Elliott Carter to Pierre Boulez, from Steve Reich to Thomas Adès, other composers couldn't have done what they did without it as inspiration. Talking many years after its composition, Stravinsky claimed he had to put himself in a kind of creative trance to compose it, an echo of the fate that befalls the poor girl who dances herself to death in the ballet's climactic Sacrificial Dance: "Very little immediate tradition lies behind The Rite of Spring – and no theory. I had only my ear to help me; I heard and I wrote what I heard. I am the vessel through which The Rite passed."

The ‘knock-kneed’ Lolitas of the original Rite of Spring

Let's deal with the riot first. For all the "heavy noises" and shouts Stravinsky says came from the auditorium, there is no evidence of mass brawling, and nobody tried to attack the dancers (although the conductor Pierre Monteux remembered that "everything available was tossed in our direction"). One critic described the whole thing as merely a "rowdy debate" between rival factions in the audience. And if the boos and hisses had been so appalling, why would Diaghilev have been as pleased as Stravinsky says he was? "After the performance," he noted, "we were excited, disgusted, and … happy. I went with Diaghilev and Nijinsky to a restaurant. Diaghilev's only comment was, 'Exactly what I wanted.' Quite probably, he had already thought about the possibility of such a scandal when I first played him the score, months before."

It would certainly be an exaggeration to say the whole thing was engineered as a publicity stunt. But how to explain the fact that the audience was protesting right from the start about something they hadn't properly heard yet? Significantly, when the score was performed in Paris for the first time as a concert piece just a year later, there were huge ovations, with Stravinsky carried on the shoulders of his fans in triumph.

It was, it seems, the wilful ugliness and lumpenness of Nijinsky's evocation of Russian prehistory that was really shocking to audiences – the "knock-kneed Lolitas" Stravinsky wrote of. The dance offended their sense of beauty and their vision of what a ballet should be, as much as if not more than the music. Anyway, at the premiere, the radicalism of Stravinsky's score could hardly be heard for cat-calls, although some reports suggest the boos had calmed down before the climax. Stravinsky had great praise for Monteux's cool head, calling the conductor as "impervious and as nerveless as a crocodile". He added: "It is still almost incredible to me that he actually brought the orchestra through to the end."

The paradox of the primitivism in The Rite is that it can be heard as both a horrifying vision of the pitilessness of nature – and as an expression of the inhumanity of the machine age. The fate of the "chosen one" in the Sacrificial Dance is particularly chilling. She is caught in an unstoppable rhythmic vortex from which there is only one way out: through the terrible dissonance that ends the piece, and the single chord that kills her. This is music that manages to sound both mechanistic and elemental, making The Rite as radical in 2013 as it was 100 years ago.

Still, for all its modernity, for all Stravinsky's insistence that the whole thing came from "what I heard" (and for all that Puccini would later call it "the work of a madman"), The Rite is rooted in musical traditions. As Bela Bartók intuited, and as musicologist Richard Taruskin has shown, many of The Rite's melodies come from folk tunes – including that opening bassoon solo, which is actually a Lithuanian wedding song. Scholars have identified more than a dozen folk references so far, but there's an even more significant tradition behind The Rite.

The work is the apotheosis of a way of thinking about music that began in the 1830s with Mikhail Glinka, the first important Russian composer, and an inpiration to Stravinsky, whose music he loved. The way The Rite moves, with its blocks of music juxtaposed next to and on top of one another like a mosaic, is prefigured not just by Stravinsky's previous ballets for Diaghilev (The Firebird and Petrushka), but in music by his teacher Rimsky-Korsakov and even Tchaikovsky. Those "new" sounds with all those dagger-like dissonances? If you look at them closely, you'll find they're all versions either of common chords stacked up on top of each other, or are built from the scales and harmonies that Rimsky-Korsakov, Mussorgsky and Debussy had already discovered. And all those jerky, jolting rhythms? They're derived, albeit distantly, from the way some of those folk tunes work.

So there's nothing so old as a musical revolution. But even if it's true that Stravinsky plundered traditions both ancient and modern to create The Rite, there's something that, finally, can't be explained away, something you should feel in your gut when you experience the piece. A century on, the truly shocking thing about The Rite is still with us, right there at its climax. A good performance will merely pulverise you. But a great one will make you feel that it's you – that it's all of us – being sacrificed by Stravinsky's spellbinding and savagely cruel music.

• The London Philharmonic Orchestra plays The Rite of Spring on 17 February at the Royal Festival Hall, London SW1, as part of The Rest Is Noise festival. Box office: 0844 875 0073.