Siegfried Sassoon once wrote a poem complaining about a concert whose audience listened to The Rite of Spring as if it were "by someone dead / like Brahms", instead of rioting and yelling abuse. Indeed, most of the great works of 20th-century modernism have become part of the canon. People may still occasionally make disobliging remarks about Picasso, say, but we are used to TS Eliot's The Waste Land – it is assimilated, and no longer regarded as an awful warning of the debased, degenerate way in which things are heading.

It's worth remembering just how radical it was. Its use of non-linear sequence, of sudden cuts from one thing to another, precedes by a year or two Eisenstein's invention of montage in the cinema. One of the few precedents for its technique is the obscure Paris: A Poem by the minor Bloomsbury figure Hope Mirrlees, published by Hogarth Press in 1920, but there is no evidence that Eliot had read it; the coincidence seems never to have cropped up during his later close friendship with Mirrlees. It is a far more controlled piece than Paris, with a far more considered prosody in each of its many sections and sub-sections. That is partly because The Waste Land had the advantage of having been edited by Ezra Pound, who tightened it up and gave it much of its focus. Paris is nonetheless worth mentioning because both Mirrlees and Eliot were doing something that was in the air. It was an attempt to find a way of doing to poetry what Picasso and Braque had done with cubism, a way of seeing things in a new way, of abandoning smoothness for truth.

What, then, did the first readers of The Waste Land see, knowing little of Eliot as a person and nothing of his private life? They saw deep anguish, partly personal and partly that of the wounded culture he had quarried so extensively – "These fragments I have shored against my ruins" - in an attempt to make things cohere. Even as accomplished a critic as Edmund Wilson took several readings before accepting that Eliot knew what he was doing when he provided the poem with notes that explained parts of its structure and system as well as elucidating some of its more obscure references.

The Waste Land replaces the assumed single voice of dramatic monologues like The Love Song of J Alfred Prufrock with a polyphony of many different speakers – déclassé European aristocrats, a neurotic woman who might be Eliot's first wife Vivienne, another distraught woman ("the hyacinth girl"), a couple of cockneys bickering in a pub, a modern Dante wandering London as if it were Limbo ("I had not thought death had undone so many"), a ragtime singer, a couple of Wagnerian tenors. The longer version, cut by Pound, Vivienne and Eliot himself, was called He do the Police in Different Voices, in reference to the small boy, Sloppy, who reads newspapers aloud to the denizens of The Six Jolly Fellowship Porters in Dickens's Our Mutual Friend. Eliot's first instinct was to see what he was doing in that light – a small boy entertaining a pub crowd with tales of murder.

This is a poem full of rape – we get the story of the abused and mutilated Philomel, who became the nightingale, and the seduced, abandoned and mad Ophelia, whose "good night, sweet ladies, good night, good night" follows on from the bickering cockneys' chatter of abortion and adultery, and the story of the typist aggressively taken by her dinner guest. It is a poem of dead fathers – Ariel's lies to Ferdinand about his supposedly dead father's "sea change"; "the king my father's wreck" – and of other losses – the drowned Phlebas stands in for Eliot's friend Verdenal, dead at the Dardanelles – and thus for all of the war dead. It is a poem in which polluted rivers, and canals by the gasworks, are the barren landscapes undone by wrongful acts and unasked questions in Arthurian legend. What was old and fine has become sinister and distorted and changed; what is new is cheap and vulgar and shoddy –"we are in rats' alley / where the dead men lost their bones". It is a poem in which sex turns to bickering to the darkest of nightmares as quickly as Sosostris can turn her tarot cards.

The poem draws on and shatters into pieces the polite culture of Eliot's cultivated youth – bits of Arthurian lore, echoes of Shakespeare and Goldsmith and Ovid – as well as less conventionally acceptable literature – a line from Baudelaire here, of de Nerval there. It draws on Christianity – the agony in the garden, the unrecognisable companion on the road to Emmaus, the allusions to St Augustine in spiritual crisis – and Buddhism, with the three-fold commands of the thunder in the last section. Yet, at best, it offers little consolation; after the seeming resolution of the commands of the thunder's precepts, it bursts out in anguish again with a line from Kyd's The Spanish Tragedy – "Hieronymo's mad againe" (Eliot will have known that Kyd was a notorious atheist, one of Marlowe's School of Night). The thunder repeats, but the call to peace at the end – "Shantih, shantih, shantih" – is perhaps the peace of exhaustion rather than acceptance. Eliot is presenting a diagnosis of his, and our, sickness, but he is not yet sure of the prescription – which is why, perhaps, The Waste Land is so great a poem.

• This footnote was added on 22 April 2014. Another version of this article was launched in error. Because of the number of comments on each piece we have left both up.