Towards the end of the Royal Academy’s superlative From Russia exhibition, two photographs show the avant-garde leader Malevich at his peak and nadir. In the first, from 1920, he is a charismatic figure surrounded by fiery-eyed acolytes as he boards a train to Moscow – a journey, metaphorically and literally, to the future of modern art. A photograph from 1935 shows the artist on his deathbed, in a claustrophobic, dingy flat hung with his famous abstract paintings which were not, by then, allowed to be shown anywhere else.

What happened to the Russian avant-garde between these years? The Royal Academy does not tell us; nor has any other exhibition ever held in Britain. But now London has an essential coda to From Russia. The Hayward Gallery’s Alexander Rodchenko: Revolution in Photography is a smaller version of a landmark retrospective from Moscow and Paris and reveals, in unforgettable images as powerful as they are poignant, the story of one of the most intriguing Russian artists, and with it a record of the Soviet revolution and its aftermath in human terms.

Born in 1891 in St Petersburg, Rodchenko was the son of a props manager at a theatre club and a washerwoman, and his life thus began in the black and white – the darkened auditorium where his father rounded up the props, his mother’s gleaming laundry – that shaped his oeuvre.

An ardent communist, he fought two battles, both of which reverberated down the 20th century and still concern us now. The first was to put the artist at the service of the people; the second to make photography rival, then replace, painting. The former took him back to an ideal of the pre-capitalist Middle Ages, when artists were craftsmen with jobs integral to society. The latter catapulted him to embrace modernity and new technology – camera, machine, electricity pylon, tower block – as medium and subject.

“I want to make completely believable photos, the kind that never existed before, pictures that are so true to life that they are life itself ... that will shock and astound people,” Rodchenko wrote. That energy and faith is concentrated in his most famous photograph, the bright-eyed, laughing 1924 portrait of Lily Brik, Mayakovsky’s muse and lover, with which this vibrant show opens. Curator Olga Sviblova calls Brik “a machine of inspiration” to the Moscow avant-garde and, across a Russia where, at the time of the revolution, 70 per cent of the population were illiterate, her youthful image became an icon of the possibility of change and hope.

From the warmth and informality of his portrait of his wife and collaborator Varvara Stepanova, smiling at him as she smokes a cigarette, to the Hockneyesque ripples of water in his depiction of his daughter taking a bath, every work revels in life-enhancing quotidian detail. In “Portrait of Mother”, elderly Olga Rodchenko, emblem of Russian endurance, peers through a single lens of her spectacles. Formal qualities – her Matisse-like decorative headscarf, its polka dots echoing the round lens which in turn encloses the eye and a circular cyst; the rhythmic lines of her scarf and eyebrows, nose and lips – as well as a distillation of everyday survival caught in a fresh moment, all lift this photograph to greatness.

As with most innovative photographers – Cartier-Bresson, Moholy-Nagy, Man Ray – Rodchenko began as a painter, and he brought to the genre the rigour of his experience as a geometric abstractionist. Severe foreshortenings, unexpected, fragmentary close-ups, theatrical composition and optical patterns balance documentary flair. The aim was to capture a new world, to construct images not usually seen by the human eye: the soaring “Fire Escape (With A Man)” and the vertiginous “Balconies, Corner of a House”, taken from the courtyard of his eight-storey apartment block; the official encased in a mesh of iron and steel in “Guard at the Shukhov Tower”, the radio-transmitting antenna for the Voice of Comintern; the lone woman carrying a child up seemingly endless stone stairs – a homage to Eisenstein’s nurse in Battleship Potemkin – in a blaze of light and shade in “Steps”. Everywhere, the thrust is upward, angles razor-sharp, textures hard and unyielding.

The first room here is devoted to the radical, ideologically powerful photomontages, used for magazines, posters and movie advertisements, with which he set out to rebuild Russia’s view of itself. “USSR Under Construction” was the title for a journal whose cover shows Mayakovsky crossed with a giant pen and manuscripts. Enormous figures in crisp suits hurtle among aeroplanes above a fracturing cubist city in “Flight”. In bold sans serif capitals, Rodchenko blocks one of the lenses of Osip Brik’s spectacles with the letters LEF in a portrait advertising the magazine Levy Front Isskustva, Leftist Front for the Arts.

Was that one blind eye, replaced by dogma, prophetic? If the first half of this show bursts with the will to construct, the second half is an analysis of how a perceptive, still-committed visionary tried to accommodate himself to the repressive 1930s. In 1932, Rodchenko’s classic “Pioneer Girl” and “Pioneer with Trumpet”, taken at a youth camp outside Moscow, appeared in the magazine Proletarian Photo. To us, these breathlessly cheerful young workers look like blueprints for Socialist Realism, but Rodchenko was immediately attacked. To achieve his bizarre angles and intense close-ups, he had had to lie down at the children’s feet, pointing his lens almost vertically, and the result looked to many people like a terrible distortion: “We are surprised that comrade Rodchenko wanted to so disfigure the young, healthy face of the Pioneer,” announced critics.

Condemned as a formalist, Rodchenko was unemployable in Moscow and went off to Karelia to record the construction of the White Sea-Baltic Canal by slave prisoners, aware that if he suggested anything other than enthusiastic workers, he would join the convicts he depicted.

Watching the revolutionary art he had helped create turn into the monster of socialist realism, he was reduced to photographing athletic displays, then, with pathos and luminous brilliance, the circus and its acrobats – returning to the black auditorium of his childhood. His final commissioned work was the climax of his circus series – a lion caged in a metal ring, which Rodchenko tilts disorientatingly: a self-portrait of his now constrained life.

Yet the virtuoso design still suggests dynamic constructions of the mind. On the reverse of a sports parade photo from 1936, Rodchenko wrote during the second world war, when he was ill and nearly destitute: “I can’t believe that one day I won’t be here any more, but all this will remain.” That hunger for “life, vital, bold, ever-changing life” colours every image here and gives a sparkling insight into the interiority of being who first fuelled avant-garde dreams and then sustained individual existence under Stalin.

‘Alexander Rodchenko: Revolution in Photography’, Hayward Gallery, London SE1, to April 27. Tel: 0871 663 2519