Carlo Carra painted his Portrait of the Poet Marinetti in 1910. The author sits at his desk, eyes blazing, chewing so hard at his cigarette that it has broken off in his mouth. He clutches his pen as if he were about to plunge it, like a dagger, into the sheet of blank paper before him. The air is full of shapes and planes, irregular wedges and diagrams of linear force. Thick red snowflakes fall – a painter’s analogues, perhaps, for the writer’s furious words.

A year or so before Carra painted his picture, Marinetti had composed his Futurist Manifesto, that incendiary bomb of early modernist aesthetics, with its machine-gun rat-tat-tat of demands and desiderata: “We will glorify war – the only true hygiene of the world ... We will destroy museums and libraries ... We will sing the great masses agitated by work ... revolutions in modern capitals ... greedy stations devouring smoking serpents ... and the slippery flight of airplanes.”

“Futurism”, at Tate Modern, celebrates the centenary of Marinetti’s infectiously vibrant tract. Its impact was most keenly felt in his native Milan, where a group of painters calling themselves the Futurists set out to depict the modern city in a manifestly Marinettian manner. Each cultivated his own corners of metropolitan experience. Carlo Carra became a dilettante of the city by night, revelling in the spectral effects created by streetlamps and neon and the headlights of onrushing trams. His Nocturne in Piazza Beccaria, of 1910, pictures the city as an enchanted blur, a parallel universe where a hundred suns blaze, turning people buildings into atomised and ghostly shadows. In Leaving the Theatre, painted in the same year, women leaving La Scala in their evening finery are incendiarised by glaring streetlamps. They look less like people than fireworks, columnar explosions of colour lurching through the night air.

Giacomo Balla was fascinated by the ceaseless movement of people and vehicles in the city, created a stutteringly “simultaneist” language of painting. Balla was influenced by the dots and dashes of Pointillist painting, but equally fascinated by the attempts of late early photographers to capture movement in successive frames. His Girl Running on a Balcony of 1912, one of few Ballas in the show, is entertainingly idiosyncratic. Broken into dots and dabs, the vestigial image of a little girl is doubled, trebled, quadrupled across the canvas. The picture resembles a mosaic aspiring to the condition of cinema.

Originally, Futurism was closely allied to anarchism. Marinetti’s call to arms against the dead weight of the past – which held such a strong appeal for Italian modern artists, so long overshadowed by the achievements of the Renaissance – was also the rallying cry of a political revolutionary. Carra’s The Funeral of the Anarchist Galli, of 1910-11, is a seething, kaleidoscopic celebration of the energies of popular protest. Workers and police clash, whirling batons multiplied in vectors of violent motion that resemble the spokes of a wheel. But the emblematic image Futurist image of social revolution remains Luigi Russolo’s The Rebellion of 1911. A horde of bright mechanomorphs storms an abstract citadel, revolution thrusting into the heart of the city along an avenue of red arrows.

The most extravagantly talented of the Futurist artists was Umberto Boccioni, whose career was cut short by the First World War. His work is richly represented. The principal coup of the exhibition is to have brought together the painter’s two States of Mind triptychs, of 1911, from New York and Milan. These are dizzyingly subtle and strange works, in which illogically combined, shadowy vestiges of visual experience – the faces of travellers, trees, a shining train’s door handle – are viewed through slashing screens of gesture. Like much of Boccioni’s work, the States of Mind have their roots in the world of Symbolist and Expressionist art. Cubism is often cited as the primary inspiration for the movement, but for much of the time it was only at the periphery of the Futurists’ attention.

The title of Tate Modern’s show is misleading, since it by no means tells the full story of Futurism. Giacomo Balla is shockingly under-represented. His paintings of electric light and Toad of Toad Hall motorcars are among the classic Futurist images, yet, inexplicably, not one such work is included. In fact there are no exhibits of any kind from after 1915, which seems bizarre given that the movement only reached its rather dark conclusion, as one of the house styles of Mussolini’s Fascism, in the 1930s and 1940s.

Instead, the decision has been taken to focus on the very early years, and then to chart Futurism’s influence in such diverse milieux as Cubist Paris, Vorticist London and pre-revolutionary Russia. As a result, the display soon comes to feel very much like a book in exhibition form – and a rather academic, obscurantist book at that. This is particularly true of sections devoted to the Cubists’ various reinterpretations of, and arguments with, their Futurist contemporaries. There are strident blasts from the Vorticists and revolutionary presentiments from several Russian painters, but no real sense of why the exhibition has chosen to travel into such contested byways of Futurist influence. It would have been wiser and more illuminating to leave out all the comparative illustrations and focus exclusively on Futurism itself, in strength, depth, and historical duration. This is a missed opportunity.