Todd Haynes’s documentary paints a jagged, revelatory portrait of the New York avant-garde scene of the 1960s.Sometime in the 1960s, John Cale, a classically trained Welsh violist with avant-garde leanings, met Lou Reed, a middle-class Jewish college dropout from Long Island who dreamed of being a rock star. Their creative partnership, encouraged by Andy Warhol and enhanced by the mercurial presence of the German model, actress and singer Nico, was the volatile bedrock of the Velvet Underground, a commercially marginal band that altered the course of popular music.The Velvet Underground story is hardly obscure, and in outline it might fit fairly neatly in the standard music-documentary template. Early struggle gives way to (relative) triumph, and then the whole thing blows up in a squall of battling egos, substance abuse and self-destructive behavior. In the aftermath life goes on, solo careers are pursued, and the survivors — fans as much as artists — look back with mellow affection on the wild and heady past, brought alive by excavated television footage.“The Velvet Underground” has some of those elements, but it’s directed by Todd Haynes, a protean filmmaker who never met a genre he couldn’t deconstruct. While not as radical as “I’m Not There,” his 2007 Bob Dylan anti-biopic, this movie is similarly committed to a skeptical, inventive reading of recent cultural history. It’s not content to tell the story in the usual way, and it finds revelation in what might have seemed familiar.Haynes doesn’t just want you to listen to the reminiscences of band members and their friends, lovers and collaborators, or to groove on vintage video of the band in action. He wants you to hear just how strange and new the Velvets sounded, to grasp, intuitively as well as analytically, where that sound came from. And also to see — to feel, to experience — the aesthetic ferment and sensory overload of mid-60s Manhattan.A lot of eloquent people are on hand to talk about what it was like. Cale and Maureen Tucker, the drummer, the two original Velvet Underground members who are still alive, share their memories, as do some of Reed’s old friends and surviving members of the Warhol circle.Their faces, shot in gentle, nostalgic, indirect light, share the screen with a rapid flow — a kinetic collage — of images. While those images sometimes document places, events and personalities — offering up Allen Ginsberg, Max’s Kansas City and a news clip about the downtown scene narrated by Barbara Walters — they serve more importantly to link the Velvets’ music to the experimental cinema of the time.From left, Paul Morrissey, Andy Warhol, Reed and Tucker in a split-screen frame from the film, which places the band in context of the aesthetic ferment of mid-60s Manhattan.Apple TV+Warhol was, along with everything else, a filmmaker, as was his associate Paul Morrissey. Haynes dedicates “The Velvet Underground” to the memory of Jonas Mekas, the great champion and gadfly of New York’s cinematic vanguard who died in 2019. In the film, Mekas marvels at the sheer abundance of artistic activity in the city in the early ’60s, and the constant blending and cross-pollination that was taking place. Traditional boundaries — between poetry and painting, high art and low, film and music, irony and earnestness — weren’t so much transgressed as shown to be irrelevant.It was a remarkable time, but not exactly a golden age. Haynes respects the art too much to idealize the artists, or to impose retrospective harmony on their dissonances. The overt cruelty and menace of the music — the droning and distortion behind lyrics about addiction, sadism and sexual exploitation — didn’t come from nowhere.The film critic Amy Taubin, who appeared in a Warhol film about “the most beautiful women in the world,” bluntly recalls that the Factory, Warhol’s headquarters, was a bad place for women, who were valued for their looks rather than their talents. An aspect of Warhol’s genius was a gift for using people, and often using them up. Reed, who died in 2013, is a posthumously beloved figure, but not many of his contemporaries would describe him as a nice person.And niceness was, in any case, antithetical to what the Velvet Underground was trying to do. “We hated that peace and love crap,” Tucker says. The artist Mary Woronov, who toured with the Velvets on the West Coast, elaborates on their hostility to the California counterculture: “We hated hippies.” Never a political band, it nonetheless articulated a powerful protest — against sentimentality, stupidity, false consciousness and positive thinking — that would sow the seeds of punk rock and later rebellions. Testimony to their influence is provided by the singer-songwriter Jonathan Richman, who estimates he saw them live 60 or 70 times when he was a teenager in Boston, and whose enthusiasm is undimmed more than half a century later.Drop a needle on any Velvet Underground record — or queue up a playlist, if that’s how you roll — and what you hear will sound new, frightening and full of possibility, even on the thousandth listen. “The Velvet Underground” will show you where that perpetual novelty came from, and connect the sonic dots with other, contemporaneous artistic eruptions. As a documentary, it’s wonderfully informative. It’s also a jagged and powerful work of art in its own right, one that turns archaeology into prophecy.The Velvet UndergroundRated R. “Heroin,” “Venus in Furs,” “Sister Ray” — you do the math. Running time: 1 hour 50 minutes. In theaters and on Apple TV+. More