Sometimes Sigmar Polke's art is celestially mysterious. He's been known to paint with leaves of gold, with beeswax and with space dirt. In 1988 he took a kilogram of meteorite and ground it into powder, which he sprinkled from above on a 13-foot-tall canvas. The skyscape that resulted smells strangely of burnt metal. It glistens as you pass it -- it's like a sea of stars.

Sometimes Polke's art is utterly banal. The 1960s pictures that commence his retrospective at the Hirshhorn Museum and Sculpture Garden are bald, deadpan depictions of ordinary objects -- plastic washtubs, doughnuts, socks. They're pop art without pop. They're like Warhols shorn of glitz.

Halfway through the German painter's vast, perplexing show is a canvas that presents a sequence of equations: "2+3=6," it says, and "3+4=9." Nor is it contradicted by the 77 other works that surround it. In the world of Sigmar Polke, nothing quite adds up.

Were Polke a composer he'd write, with equal ease, for harps of hammered gold and two-penny tin kazoos. Listen and you'll hear shamans' chants, and Muzak, delicious French gavottes and harsh, sardonic laughter in the music of his show.

Polke, like his eerily disjunctive art, is insistently elusive. He does not have a dealer. When he speaks, he speaks in code. His odd layering of images -- his willingness to paint, say, a mushroom on a soccer game or a goose over a deck chair -- is said to have inspired the work of David Salle, Julian Schnabel and other '80s hotshots, though where they saw their Polkes remains something of a mystery. The painter, who is 50, has shown rarely in America. The Hirshhorn's retrospective -- which was organized last year by the San Francisco Museum of Modern Art -- is his first big touring U.S. show.

"The present exhibition," the critic Peter Schjeldahl tells us in the catalogue, "is a crash-course chance to alleviate our ignorance. But anyone who thinks that simple familiarity with Polke's output will dispel its obscurity has another think coming. To learn more and more about him ... is to know less and less. His art is like Lewis Carroll's Wonderland rabbit hole, entrance to a realm of spiraling perplexities, one of which is his uncanny relation to American art, first as a provincial follower, and later as a seminal influence."

Polke doesn't draw that well. He doesn't really try. He frequently appropriates -- he borrows gladly from cheap magazines and as gladly from the masters, from Durer, say, or Goya, Dali or Picasso. All of this, by now, is thoroughly familiar, but every time you try to place him in some neat postmodern cubbyhole, he darts off into regions that are his alone.

The first work here encountered is a suite of little paintings whose title is "The Fifties." Anyone who's old enough to call to mind that dull decade -- its pink shirts and black neckties, or its corny South Seas restaurants, all plastic palms and black-lit moons and painted bamboo staves -- will recognize their colors and their mood of kitschy glamour. But the vast, strange works that end the show aren't like that at all. Imageless abstractions, alchemical in spirit, they're painted with the strangest stuff -- with that granulated meteor, with tellurium and silver. One bears on its surface neolithic knives. A second, says its label, is of "silver nitrate painted on invisible hermetic structure." Blank white when it was sold in 1988, that arcane and ghostly canvas has since begun to darken, gradually developing like some strange slow-motion Polaroid. It's as if it were alive.

So who is Sigmar Polke? Is he goof or wizard, a dullard or a sage?

The strangest thing about his show is that despite its ceaseless shiftings -- its messes and precisions, its prayers and sour tauntings -- it somehow holds together. If he could paint on air he would, until all his layered thoughts, his memories and prophecies, his inventions and quotations, image after image, would be at once apparent. His chief American competitors prefer a look of blankness. In Schnabel's pompous paintings, and in Salle's too, the pieces seldom fit. You will never understand what it is I'm getting at is the message they deliver. Polke isn't like that. He's more honest, more confessional. Wandering his show is like walking through his mind.

One of his best paintings, "Stairwell" (1982), lives permanently in Washington. James Demetrion, director of the Hirshhorn, purchased it some years ago for his gallery's collection. Like many other Polkes, it is painted not on canvas, but on purchased figured fabric. It is dense with innuendo, and with memories of movies. To read it is to dream.

It stars a faceless man of action, a spy or a detective. The splashed colors that surround him are shadowy as Bogart films or Technicolor bright. He's sneaking up a creaking stair, a pistol in his hand. And he's accompanied by ghosts. One -- a quote from Aubrey Beardsley's drawing of Camille -- floats through this odd thriller like some preposterously overdressed perfumed femme fatale. The white-eyed, white-toothed purple splodge at the picture's upper left is a figment of that evil that, as the Shadow knows, lurks in the hearts of men. This corny, spooky painting smells of satire and menace, of stale popcorn and James Bond.

Polke's art is rarely gentle. It's often fiercely impolite. The mocking painting he has titled "Higher Powers Command: Paint the Upper Right Corner Black!" sneers at Ellsworth Kelly, and even at Malevich. He also boos the great Picasso. Though Polke can paint beautifully, he's unafraid of ugliness, of hurled paint and of smearings. One of his best pictures, "Scissors" (1982), depicts a Polish medium in trance. She is levitating metal. Beneath her outspread fingers a pair of heavy scissors are floating in midair. Is she -- is Sigmar Polke -- a psychic, a magician or a Yuri Geller trickster? An odd mood of chicanery is felt throughout his show.

His politics are suspect too. The 14-foot-high, eight-foot-wide painting he calls "Camp" (1982) -- all darkness and barbed wire, it was painted on cheap blankets with the aid of burning coals -- seems an open condemnation of the death camps of the Nazis, or of the Iron Curtain. But the painter will not have us comfortably accept him as a good, guilt-ridden German. The swastikas that dance through Polke's "Paganini," a picture also painted in 1982, have been drained of their horrificness. Playful, half-innocuous, they're like tiny tinkling chimes.

The severed heads on spikes in Polke's "Liberty, Equality, Fraternity" (1988), and the hanged men and dead soldiers that he has borrowed whole from 18th-century images of the French Revolution, suggest a deep disgust with violence -- until one feels again the delight in overthrowing, and in revolution, that smolders in this show.

Much German art, both old and new, is dense with glints of death, with suffering and monsters. The painter's armored knights, his poisonous concoctions, his insults and destructions, are often truly scary. When Polke looks toward France, he sees not sun but severed heads. When he honors Goya, he does so for the Spaniard's owls and grinning witches. Polke gnaws his viewers. His art is meant to be disturbing. It is his cursing, his derision, his mocking erudition -- and his rattling of our confidence -- that unifies his show.

"In his work," writes Schjeldahl, "nothing is either quite subjective or quite objective, though everything seems ordered by some mastering comprehension. It must be said that this comprehension -- the Polkean daemon -- is extremely unreassuring. It is derisive, impudent. Mephistophelean. It steers awfully close to outright cynicism. It is without apparent pride, a rascally character. It has an affinity for the debased in imagery, materials and procedures, a debasement it is at no pains to redeem -- or to not redeem, for Polke on occasion will make things that are immensely fine, with the same seemingly distracted nonchalance with which he uncorks outrages. The dynamic mess of many of Polke's paintings suggests the work of a blind man with good luck. I have a fantasy of him: he enters a pristine studio full of exquisite materials all in order, and wrecks the place. The wreckage -- a series of paintings -- is removed. The studio is bulldozed."

But painting is resilient. It has a way of growing through attacks on its conventions. When looking at a Polke -- at those dark satanic stains, those vandalizing borrowings, those odd cartoony scrawls -- you are not quite sure you get it. His enigmatic work is wrapped round with perversities. Yet when experiencing his paintings you do not doubt his force, or his love- and hate-filled loyalties to the act of making art.

Polke is known to be upset with the Hirshhorn's installation, most of which, it turns out, he designed himself. The artist chose on Wednesday to stay away from a Hirshhorn birthday dinner given in his honor, but then he is well known for squabbling with museums and collectors. His show remains impressive. And just right for the Hirshhorn. In recent years it's brought us, among other major shows, Lucian Freud's and Francis Bacon's retrospectives, fine modern British sculpture, Giacometti's too, and a touring show of pictures by Polke's one-time painting pal, the German Gerhard Richter. No American institution has shown us more important recent European art than James Demetrion's museum.

Polke's exhibition is supported, appropriately, by the Andy Warhol Foundation for the Visual Arts. It has also received funding from the National Endowment for the Arts. It will travel to the Museum of Contemporary Art, Chicago, and to the Brooklyn Museum after closing here May 7.