A fast and furious chase, full of physical gags and gangsters, with jokes at the expense of American imperialism. A hallucinatory horror, where ordinary objects take on a life of their own, scripted by a literary theorist. A bed-hopping love triangle, simmering in a cramped flat. A big-budget science fiction spectacular, full of futuristic sets and bizarre, revealing costumes. A workers' strike, depicted via special effects and pratfalls. A film about film-making itself, with no plot, no words, no narrative, which is somehow the most thrilling film you'll ever see. A film about collective farming with full-frontal nudity and inscrutable, poetic metaphors. A film about mutinous sailors that manages to accidentally invent the action film as we know it.

This is Soviet cinema in the 1920s. An almost entirely state-run cinema, devoted to propagating communist doctrine by the most nakedly propagandistic means, and subject to heavy intervention from the Soviet bureaucracy. One might suspect a Soviet, socialist cinema to be a grimly bureaucratic thing itself, a jargon-laden matter of boy meets tractor, devoid of excitement or drama, with everything subordinated to the political message. Or you might expect a school of film that repudiates the commercial, blockbuster cinema in favour of didactic accounts of thesis, antithesis and synthesis. Yet although numerous directors talked of the "dialectical film", it was never any such thing. You might expect something humourless and leaden, only to find films which resemble Buster Keaton far more than they anticipate Jean-Luc Godard. Soviet film in its first decade and a half managed to set the pace for world cinema, and its formal innovations are still being digested 80 years later.

The Russian Film Pioneers section of the British Film Institute's Kino season, running until 30 June, is a rare chance to explore this seemingly paradoxical world, where a series of still extraordinarily watchable popular films were made wholly without the intervention of the market. What is especially striking about them is how little the Soviet films of the 1920s resemble the common models of left-wing, activist (or any kind of "art-house") cinema. There's not the slightest hint of the worthy, socially engaged realism of Ken Loach and those he has inspired – the characters in these films are cartoons, and the scenarios are often fantastical, joyously so. And although Godard invoked the Soviet heritage when he started making collective films in response to May 1968 as the Dziga Vertov Group, there's nothing further from his brackish, deliberately stilted and oppressive approach to political cinema than the fast-paced, panoramic, sweeping, playful Bolshevik documentaries of Dziga Vertov himself. For films that are so influential on filmic technique, especially in their development of fast-cut montage, they have had surprisingly few direct successors.

One reason for this is the very specific political context of these films. Lenin's claim that "for us, the cinema is the most important of the arts" is often quoted, but the Soviet film industry didn't really take off until after his death, aided by the economic recovery from the first world war and the 1918-21 civil war. And it's during the civil war that many of these film-makers, later inaccurately portrayed as aesthetes out of their depth in politics, began their work. Sergei Eisenstein, Lev Kuleshov, Dziga Vertov – all had been involved on the Bolshevik side in this complex, savage war. Vertov's earliest newsreel series, the self-explanatorily named Kino-Pravda, even contained one reel supportively depicting Lenin and Trotsky's 1921 suppression of the Kronstadt sailors' mutiny. These were not dizzy idealists caught up accidentally in politics, but political thinkers willing to lend their expertise to the Bolshevik cause. They all faced heavy criticism and were often banned in the 1930s – but it was the regime that had changed its priorities, not the film-makers.

Their attempts at Bolshevik propaganda film mirrored the contradictions and aspirations of Bolshevism itself. In its early years, the Soviet Union aspired to become a kind of "better America", where a then 80% rural country would apply the techniques of Henry Ford, Thomas Edison and Frederick Wilmslow Taylor, combine them with workers' self-rule and Marxist "social science" and create a new and superior society in the process. In the end, neither Fordism nor workers' democracy had much role in the USSR, but both were fundamental to Soviet film-makers.

Sergei Eisenstein used non-actors and made the crowd his hero; Vsevolod Pudovkin, in Mother and The End of St Petersburg, depicted revolution from below, without a leader in sight. But both of them were as infatuated with American popular entertainment – Charlie Chaplin, Harold Lloyd, Douglas Fairbanks, a landscape of skyscrapers and jazz – as they were with American technology. The final, and perhaps strangest ingredient was formalist literary theory, a seemingly apolitical school of textual analysis founded in the late 1910s by Viktor Shklovsky (who fought against the Bolsheviks as a member of the Socialist Revolutionary party) and Osip Brik (who fought for them, serving in the embryonic KGB, the Cheka). Formalism, with its fixation on the strangeness of objects and the transformational qualities of the literary "device", was key to early Soviet cinema's aim to revolutionise perception – and both Shklovsky and Brik wrote film scenarios. Imagine a parallel universe in which Jacques Derrida and Michel Foucault wrote adventure films and slapstick comedies in their spare time.

The milieu the directors came from was equally peculiar – on the one hand constructivism, the utopian-utilitarian twist on the warlike Italian futurist movement, and on the other, Proletcult, an association formed by philosopher, experimental "scientist" and science-fiction novelist Alexander Bogdanov that favoured the development of a uniquely proletarian culture. The son of an architect, Eisenstein rather improbably made his early films under Proletcult's aegis, and all the directors had some involvement with the futurist magazine LEF. Somehow, the films of the 1920s managed to contain this combustible mix of theory, populism and politics. One of the most impressive of the early syntheses was a group called the Theatre of the Eccentric Actor, which published hysterical manifestos hailing the "arse of Charlie Chaplin" and condemning to death the realist, high-society, "theatrical" film, proposing instead a combination of the circus and constructivism.

The first of these directors to secure (meagre) funding to make a film was Lev Kuleshov, who had spent years researching movies, along with a studio of self-proclaimed "failed actors". His investigations led to a recognition of the sheer speed of American movies and the imperative to produce "the maximum of entertainment". The resultant film, The Adventures of Mr West in the Land of the Bolsheviks, led by the awkward, charismatic talents of his wife Alexandra Khokhlova, was important for Eisenstein's debut work.

Strike was part of a series of films commissioned to commemorate the failed "first Russian Revolution" of 1905. Historical accuracy was of little importance to Eisenstein, who favoured instead acrobatics, trick effects and a final, bloody and cathartic (or rather, orgiastic) battle between the strikers and the police.

Kuleshov's "failed actors", among them the former boxer Boris Barnet, then struck out on their own, producing highly Americanised satires on the remnants of the bourgeoisie in Soviet life. And finally, the Theatre of the Eccentric Actor got into film; in its wildest work, The New Babylon, the ostensible topic was again the defeated revolutions of the past, this time the Paris Commune of 1871 – the method was avant-garde slapstick, all can-cans and barricade fighting.

No doubt these historical liberties set a dubious precedent. LEF criticised Eisenstein's October (1928) for distorting the record of the Bolshevik revolution, with the lead actor a non-professional who looked, they sneered, like "a statue of Lenin". Reconciling the need to proselytise, to entertain and to experiment was seldom easy, and with exceptions such as Mother (at home) and Battleship Potemkin (abroad), Soviet films never matched American movies at the box office. In the early 1930s, the Party got round this simply by banning imports.

Perhaps the only Soviet director who was not intent on creating a socialist spin on populist, jazz age America was Dziga Vertov, who disdained any connection with commercial film whatsoever. His documentaries still retain a thrilling velocity which was surely taken up from American comedies and the epics of DW Griffith (a favourite of Lenin's). His finest film, Man With a Movie Camera, which has started to get the attention it deserves over the last two decades and is now one of the most-seen Soviet films, was too extreme even for the Soviets in its abandoning of all conventional structure in favour of a combined city-portrait and delirious meditation on the cinema itself. Somewhat more obscure is the follow-up, Enthusiasm, where Vertov takes a similarly hard-line approach to sound. Charting the mechanisation of the Donbas area of the Ukraine, it is an aural assault of collaged industrial sounds, as harsh, heavy and blaring as the work itself. And with the first Five Year Plan to industrialise Russia and the cementing of dictatorship, Vertov's relentless attention to the human difficulties of industrial labour was no longer useful. Donbas workers praised his film for showing how hard their lives were; for the Party, that was exactly the reason why it was suspicious.

The Soviet directors' populist futurism would barely last more than a year or two into the 1930s, but their innovations survived. In the USSR, Stalinist musicals such as Grigory Alexandrov's once-famous Volga-Volga (1938), heavily formed by Stalin's own interventions, were as anti-realist and "eccentric" as their forbears. The difference was that the politics was actually relegated to the background, that the propaganda was less obvious, not more; until the final scene and its Soviet "message", the film could be enjoyed as a straightforward, silly musical about rural life. From the 1960s onwards, tributes to Eisenstein were obligatory for the American "movie brats", references to Vertov equally mandatory for European experimenters. Yet while the politics disappeared in the former, the Soviet imperative to provide "the maximum of entertainment" disappeared in the latter. For a brief, intense flash between 1924 and 1930, "high" and "low" art – Marxist theory and the Keystone Kops – came together. Political film has seldom been as explosive since.

Kino: Pioneers of Russian Film is at BFI Southbank, Belvedere Road, London SE1 until 30 June. www.bfi.org.uk/kino.html