If you had asked me 30 years ago what sort of career Anish Kapoor was going to have, I would have replied: a short one. Not because I am a bad caller of these things, and a lousy art critic, but because Kapoor’s early sculptures felt rather derivative. Providing you had been to India, that is.

Kapoor and I are more or less the same age. I remember his debut vividly. The first works of his to have an impact were bright heaps of unmixed pigment — red, yellow, blue — deposited on the gallery floor in Jungian clusters and looking as if they had been tipped out of giant cake moulds. The intensity of those unmixed colours gave away their Indian origins. Anyone who has ever approached a Hindu temple will recognise these startling hues from the stalls of the pigment pedlars lining the final mile. In India, temple stall after temple stall offers a Kapoor experience in miniature.

My mistake, and of course it was a huge one, was to imagine that quoting from his origins was all Kapoor would ever seek to do. It’s not that I did not respond to his unmixed pigments; the sight of them was electrifying, then and now. But, like one of those Booker Prize winners who writes a fine novel about their childhood in Calcutta, and that’s it, I thought he might not have a volume two in him, that his subtext would become his text. How wrong was I?

A selection of Kapoor’s colour shockers pops up in the first room of the Royal Academy’s impressive half-retrospective of his career so far. It’s half a retrospective because the RA is too small to accommodate the full beast, and because Kapoor is too alert and ambitious a sculptor to settle for nostalgia. When the RA invited him into its galleries, it invited him into a new range of sculptural possibilities, which he explores here with characteristic fierceness. This is a battle as much as it is a retrospective: Kapoor v the Royal Academy’s spaces. Why, there’s even a cannon in the show, firing splats of gooey Napoleonic wax at the gallery walls. Extraordinary.

The battle actually commences outdoors, where a column of giant silver balloons soars up from the RA’s courtyard as if set free by a party of very large schoolkids. Each of the silver balloons flashes a distorted reflection of the surrounding courtyard at us, creating a wonky totem pole of jumbled Academy vistas. It’s a bubbly and playful sight that seems to promise all the fun of the fair in the experience ahead. Kapoor has a telling talent for reminding our bodies of experiences they recognise from elsewhere. My guess is the inspiration for the courtyard piece was nothing more complicated than a day out with his children, buying helium balloons.

Having lured us in with the promise of fun and games, the show spends the rest of its length growing darker, weirder, weightier, as it digs ever deeper into our memory store of sculptural effects. Seeing the pigment pieces again, I was struck this time by how thoroughly irrelevant their origins are. What counts is not what triggered them, but what they trigger in you. Kapoor’s startling colours assault the eyes with a force that is closer to rape than to persuasion. Although we live in a coloured world, most of the colour we encounter is hushed, toned down, veiled. Not this stuff. These reds are to most reds what the roar of a Ferrari is to most car noises.

So fierce is their impact that, at first, you will not notice the most compelling sculpture in the room, a gentle white swelling on the wall called Mother as Mountain. The swelling has no explanation around it, no context. It’s just a wall behaving strangely, as if something precious were growing inside it. Your instincts sense its meaning before your brain does. It’s a sculpture about pregnancy.

Sculpture is, of course, uniquely able to go this deep. It speaks to us on our most basic level: body to body, matter to matter. When it has finished toying with our memory store, the show begins drilling down to our next level, the subconscious, where it starts probing indelicately in the psycho­sexual arena. How else to understand the appearance here of an immensely phallic cannon, firing wads of red wax through an adjacent doorway onto the wall ahead? If the RA had a porn alarm, this thing would set it off. The ejaculating brute even has testicular-shaped canisters of compressed oxygen dangling from its loading end. The noise when it explodes is terrifying. Sculpturally, I cannot imagine a more crudely masculine spectacle than this brutal splashing of wads of sperm across the pristine gallery surfaces. The crowds will love it. It’s a phenomenal piece. We are in the presence of a very effective gallery operator.

These days, it’s a toss-up whether Kapoor or Antony Gormley is Britain’s most successful public sculptor. Both are able to work on a daunting scale that would frighten lesser sculptors. Both have become masters of sexy audience manipulation of the kind that art has increasingly begun to demand since it joined the entertainment industry. The central room here, full of twisted aluminium surfaces crowded with madly distorted reflections, like the hall of mirrors at a funfair, is simply an irresistible experience. Is there a human being on earth who does not enjoy the sight of their own reflection turned upside down and gurning crazily? Kapoor knows there is a monkey in us all. What I admire about him most, however, is the unwavering depth of the experiences he conjures up. Initially, the bendy mirrors are merely fun to look into, but soon enough their relentless distortion turns the innocent fun into serious reality questioning. Likewise in the room before, where a wall of yellow pigment envelopes you and makes you feel cosmic — then pinned, then helpless, then insignificant, as if you are being sucked up by a giant vacuum cleaner.

Everything here that seems gentle at first begins biting as soon as your back is turned. The show’s most daring and ambitious sculpture is an indescribably strange thing: a huge wagon of red wax, about the size of a suburban bungalow, that progresses slowly along the entire length of the galleries on a mechanised rail. As the monster squeezes itself through the various gaps, messy blobs of its surface are scraped off onto the surrounding walls. When it finally emerges in the centre of the room, the snail-paced behemoth feels as brutally overscaled as a train in a kitchen. But the bit of its journey I enjoyed best was when it slid slowly and tightly through the doorways, and went all phallic on us, and thrusting, and psychosexual. The biggest penis in the world was shagging the Royal Academy. That’s how weird a sculpture it is. That’s how good a show it is.

Anish Kapoor, Royal Academy, W1, until December 11

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