On May 4, 1963, the Rolling Stones, then a scrappy quintet known mostly for banging out Chuck Berry covers, gathered for their first official photo shoot on the streets of London’s Chelsea district.

Five bad boys in the making, some still sporting adolescent pimples, they slouched in ratty sweaters, rumpled jackets, and ill-fitting trousers, looking like students stumbling through a three-day bender after getting expelled.

“Word got out that the results of the session were disgusting,’’ Andrew Loog Oldham, the Stones’ manager, later said. And he could barely contain his glee. That just-got-out-of-bed, to-hell-with-you look, Oldham added (he used considerably saltier language) “would define them and divine them.’’

But it would hardly limit them. Over the next five decades, the Stones would turn the stage into the world’s largest runway, transforming their look constantly and radically, even as they stayed true to their filthy, blues-based sound.

The band’s vast fashion legacy is on full display at a show opening Saturday in New York, “Exhibitionism — The Rolling Stones,’’ billed as the largest collection of the Stones’ stage outfits, musical instruments, and memorabilia ever assembled. The show takes place at Industria, a sprawling studio and event space in the West Village, after a five-month run at the Saatchi Gallery in London that drew more than 350,000 visitors.

The retrospective, curated by Ileen Gallagher, formerly of the Rock and Roll Hall of Fame, digs deep from collectors’ vaults and the band members’ closets. The black-and-red Lucifer cape that Mick Jagger wore at Altamont? It’s there. The antique toy drum kit that Charlie Watts used for “Street Fighting Man’’? It’s there, too.

The techy, multimedia exhibition contains old diaries, notebooks filled with lyrics, original cover art, and historically significant guitars in glass cases. Video screens flicker with historical concert footage, and recorded audio of the band talking about songwriting wafts overhead.

The exhibition also features meticulous re-creations of a Stones recording studio and the infamous hovel at 102 Edith Grove in London that Jagger shared with Keith Richards and Brian Jones in 1962 and ’63, complete with peeling wallpaper, discarded Playboy magazines, and an artificial odor that suggests stale beer and old socks. (“I kept saying, ‘C’mon guys, there are too many beer bottles, too many dishes in the sink,’’ recalled Jagger, in a telephone interview from Los Angeles last week.) Think of “Exhibitionism’’ as the King Tut exhibit for the dad-jeans set.

But it is the floors filled with period Stones outfits, many not seen in public for decades, that serve to underscore the band’s influence on rock style.

Although David Bowie generally got credit for being rock’s ultimate chameleon, the Stones went one further: They invented the very image of the modern rock star.

With a fashion sensibility that was one part Vogue magazine and one part skin magazine, the Stones laid the groundwork for punk, dandified Mod, pushed psychedelia to its cartoon extreme, and basically invented glam rock. And that was just the ’60s.

“You want to be new; you want to be eye-catching and yet elegant, but yet crazy because you’re onstage,’’ said Jagger, explaining his sartorial philosophy in the interview. “It’s not just five blokes in blue jeans going on with a lot of amps, you know what I mean?’’

The collection starts at the beginning, with Jones’s houndstooth-check jacket from 1963, a relic of that moment where the Stones attempted to be G-rated teen idols in matching uniforms, à la Herman’s Hermits.

“In the early ’60s, one of the big fashion things was the Beatles’ jackets,’’ Jagger said. “That was something different. Clothes were immediately talked about on men, as well as on women performers.’’

Even so, he added, the band was always mystified why their manager, who curated its bad-boy image as the anti-Beatles, wanted to experiment with uniforms. “It was a really weird thing, the matching jackets,’’ Jagger said, because Oldham’s “whole thing was to be not like that. He was the one who wanted to be different.’’

And very quickly, they were. By the time the Stones were starting to cross over in America in 1964 and 1965, they already looked vaguely dangerous, at least to teen-pop audiences weaned on Neil Sedaka.

“You saw the scruffiness, the down-dressing that really didn’t exist in the American vocabulary: the mismatched look, the leather jackets, adopting some of the traditional rhythm-and-blues style,’’ said designer Anna Sui, a lifelong Stones fan. “And then, throwing in a pair of white shoes. It was just like, ‘Wow, what is this?’ That’s how guys reacted. And to this day, you see guys dressing exactly that way.’’

Soon, the Stones were serving as global ambassadors of a very different style, the Swinging London dandy look coming out of Carnaby Street and King’s Road. That is exemplified in the exhibition by Watts’s blue-and-green tartan suit by Granny Takes a Trip, the seminal King’s Road boutique of the era, and Jagger’s red Grenadier military guardsman drummer’s jacket, which he wore while performing “Paint It Black’’ on the TV show “Ready Steady Go!’’ on May 27, 1966.

Victorian cravats and Regency-era ruffled shirts were the order of the day. “It was really the ‘anything goes’ period,’’ Jagger said. “There was a lot of antique clothing being sold, and a lot of it was revived Romantic stuff in velvets and things like that.’’

That ’60s dandy look may have borrowed from the gender-blurring Oscar Wilde style vernacular. But it was scarcely a hint of what was to come.

On July 5, 1969, during a memorial concert in London’s Hyde Park for Jones (the band’s founder who drowned shortly after being fired from the band), Jagger strode onstage in white voile Michael Fish man-dress with ruffled bishop’s sleeves and a bow-laced front. A re-creation of the dress is on display in “Exhibitionism.’’

By the landmark 1972 tour, it was simply expected that Jagger would take the stage in eye-shadow and skintight, velvet jumpsuits by Ossie Clark, three of which are on display.

By the ’90s, the Stones were a Fortune 500-level touring behemoth, and their clothes reflected their status as the new establishment.

After 54 years and seemingly 54,000 fashion experiments, the only remaining question seems to be whether Jagger regretted any particular outfit. “Ah, that’s a horrible question!’’ he said with a laugh. “You’re bound to make mistakes.’’