The Museum of Modern Art has never known quite what to do with Willem de Kooning. You can package Jackson Pollock as drips and Barnett Newman as zips, but de Kooning, who painted both opulent abstractions and big, blowsy dames, resists easy branding. So, apart from a show of late work in 1997, the museum has neglected him, until now.With the opening of “De Kooning: A Retrospective” this coming Sunday, decades of overdue debt are paid in full. The show, which fills MoMA’s entire sixth floor with some 200 paintings, drawings and sculptures, is exhaustively comprehensive, exhaustingly large and predictably awe-inspiring. It not only positions de Kooning far forward in a 20th-century American cavalcade of stars, it turns his career into a kind of Rose Bowl float of creative exuberance and invention.Most usefully, the show lets de Kooning be complicated: it presents his art as a bifurcated yet unitary phenomenon. John Elderfield, chief curator emeritus of painting and sculpture at MoMA, unfurls seven decades of work through as many galleries. In each we see abstract and figurative paintings of roughly the same date not just side by side but interacting, intertwining, merging identities; abstraction breathes, smiles and bleeds; figures shimmer and shudder apart into color and line.In its shape the show adheres to a classic cradle-to-grave survey model, beginning with a still life that de Kooning painted at the age of 12 in his Dutch hometown, Rotterdam. By his early teenage years he was already working in a commercial design firm, where he learned sign-lettering and paste-up techniques like tracing, copying and layering.All of this came in handy while looking for work when he first arrived, after stowing away on a freighter, in the United States in 1926. More important in the long run, his design training, with its emphasis on creating an organic-looking whole from many pieces, ended up as the formal bedrock of his art.He was lucky in being, by temperament, chronically hungry and omnivorous. In the first two sections of the show, which take us in giant steps up to 1949, we see him devouring his way through visual history past and present, gobbling up images from Ingres, Rubens, Soutine and Picasso; from contemporaries like Arshile Gorky; and from movie ads, Sunday comics and the graphics in New York police gazettes.While some artists and thinkers of the day were promoting an art of utopian purity, one that required shutting a door between art and life, de Kooning’s appetite took him in the opposite, though really no less utopian, direction. He wanted to open everything up, to bring — to squeeze — everything into art: high, low; old, new; savagery, grace.And so he did, in a laborious, pieced-together, piled-up, revision-intensive way. Far from being the sort of impulsive, gut-spilling artist implied by the term “action painting,” he was a deliberator. Every painting was a controlled experiment.Typically he would start with a drawing, add paint, draw on top of the paint, scrape the surface down, draw more images traced and transferred from elsewhere, add paint to them, and on and on. Given this process, it seems astonishing that he was so prolific, until you remember that he was virtually never not working, trying this, tweaking that, scrapping failures, starting afresh.Whenever the enormousness of the MoMA show gets you down, stop in front of one picture, almost any one, and linger. The basic, energy-generating dynamic of de Kooning’s art operates in microcosm in nearly every single thing he did after the mid-1940s.Experiencing the physical mechanics of his art close up, in action, is the real thrill of the show. To engage with it is to meet a person rather than just visit a monument.The late 1940s was when de Kooning first caught fire, when abstraction and figures first merged. That’s the story of “Pink Angels” from around 1945, the final painting in his first series devoted to images of women. From painting to painting, the single seated figure in the series grows less naturalistic, begins to lose its contours, to dissolve into its surroundings.