At the climax of the song “Memory,” Mo Rocca would bite a couch pillow.

“When she would go dadadadadada dadadadadada ‘Touuuch meee!’ ” he explains. It was too exciting to bear.

It was 1983, and 14-year-old Rocca would go over to his friend Mario’s house in Bethesda, Md., to listen to the Broadway cast recording of “Cats.”

When he saw the show in New York, “Memory” was thrilling, and he got an autograph from its singer Betty Buckley at the stage door. He mailed letters to the entire cast. He and Mario found a Manhattan phone book and tried calling them. He put Buckley on the cover of a fake Time magazine.

Rocca — who is now a CBS correspondent and the creator of a book and podcast called “Mobituaries” — liked seeing the actors come into the audience and sit in people’s laps. “The cats were just within reach, and that made it tantalizing,” he says. “I just think anyone who got to wear a tail and then do fouettés and roundoff back handsprings — it was just enough to make me explode.”

A few years later, Rocca was an usher for the “Cats” tour in Washington, and the mystique started to fade. “I’ve realized that it’s sort of a musical for kids.”

Not everyone is Mo Rocca, clearly, but “Cats” led many fans to bite their proverbial couch pillows, as its 18-year run was the longest in Broadway history at the time, and its productions worldwide have earned $3.5 billion. The feature film adaptation finally arrived this week, now that we have long forgotten how a show about “Jellicle cats” competing to ascend to the “Heaviside layer” on a hydraulic tire somehow became the “Hamilton” of 1982.

The Andrew Lloyd Webber-composed meowsical based on T.S. Eliot poems turned “dancing cats” into a punchline, culminating in the Internet’s baffled reaction to the mewvie’s first trailer earlier this year. What exactly made the show so pawpular in the first place? How did a project expected to fall on its head somehow land on its feet? How much more “Cats” wordplay do we really want to . . . milk?

The story of the original London production has too many twists for even the hoariest backstage farce, and helps explain the fortuitous alchemy that made this thing work. The cast rehearses without any real script. The set is literally a pile of garbage. The characters have names such as Carbucketty and Bustopher Jones, something out of a toddler’s stuffed-animal pageant. The composer has to take a second mortgage on his house.

For some reason, the head of the Royal Shakespeare Company gets hired to direct, and he also stays up at night adapting another Eliot poem into the lyrics for the most notorious earworm in shopping mall history. He is moonlighting, all alone, as a writer of “all alone in the moonlight.”

But the leading lady who sings it (Judi Dench!) snaps her Achilles’ tendon and gets replaced. Days before the first performance, the composer and producer tell the director they are pulling the plug, but he ignores them. On opening night, the theater gets a bomb threat. But the show earns raves from London critics, and it makes everyone famous and wealthy.

“It was this living, breathing, moving piece of art, almost like an art installation at some major museum,” recalls Buckley, who played Grizabella when the show opened on Broadway a year later (to more mixed reviews).

Her disheveled, ostracized “glamour cat” came from an Eliot poem that was never published because it was too sad for children, and the piece ended up as the musical’s emotional center. Despite the weirdness of “Cats,” its story is similar to Broadway’s previous longest-running show, 1975’s “A Chorus Line,” as characters take turns singing to win a contest — and the sympathy vote goes to one who is past her prime, longing for another chance.

Buckley got inspiration for her character by following homeless women on the street. “She’s a metaphor for that which we as a culture reject — aging, death, dying, loss of ability,” she says.

Christine Bogle, 49, an artist in San Diego, saw “Cats” at age 16 — the first of an estimated 115 times. Even then, she says, “I related to Grizabella very much,” as the character helped her overcome her outcast status and the knee injuries that scuttled her dancing aspirations. “At the end she’s picked to get a new life and redemption. . . . There’s something pseudo-spiritual about it.”

Bogle owns hundreds of props and other memorabilia, including a fan-made Rum Tum Tugger action figure. She was enchanted by the show’s punk-inspired wardrobe of leotards and leg warmers (and created her own costumes that she lends to productions). Not to mention the bouncy, squirmy choreography by Gillian Lynne, and the junkyard set, with its oversized shoe and can of baked beans to portray a cat’s point of view.

In his New York Times review of “Cats,” Frank Rich wrote that despite its flaws, the “reason people will hunger to see ‘Cats’ ” is that “it’s a musical that transports the audience into a complete fantasy world that could only exist in the theater.”

On the business side, everything that was “wrong” with “Cats” became right. No stars? Cast changes are not a problem. Thin story? Foreign tourists will not be lost. No money quote from the Times review? The poster can just have the yellow cat eyes and a catchy tagline, “Now and forever.”

The show created a model for the Cameron Mackintosh-produced megamusicals “Les Miserables,” “The Phantom of the Opera” and “Miss Saigon,” all with spectacle set pieces (tire, barricade, chandelier, helicopter) and iconic logos (eyes, girl, mask, sun). It also came just a few years after Steven Spielberg’s “Jaws” and George Lucas’s “Star Wars” brought an equivalent blockbuster sensibility to film.

But not all properties can easily leap between media — the “Les Mis” film was respectable onscreen, while “Phantom” was perceived as middling.

“It’s not fun to watch a great magician on television. You have stagecraft,” Rich says now. “There are things in the theater that are very hard to translate to film, just as there’s not been a successful movie of ‘Our Town.’ ”

Lloyd Webber has said that the show plays to 50 percent of the world, and that is enough. But much of the other 50 percent adamantly opposes its existence. Bart Simpson sees it and spits at the actors through a straw. David Letterman made fun of it for years, interpreting the “Memory” lyric as “midnight, and the kitties are sleeping.”

“I just hated every minute of it,” says the late Broadway legend Elaine Stritch in a biography about her, Alexandra Jacobs’s “Still Here.” As the cats came into the audience, Stritch yelled, “Don’t touch me.” She left at intermission.

A few prominent plays of the early 1990s reference it to get a laugh, including “Angels in America” and “Six Degrees of Separation.” (“Aeschylus did not invent theater to have it end up a bunch of chorus kids wondering which of them will go to Kitty Kat Heaven.”)

In Paul Rudnick’s “Jeffrey,” a “Cats” performer hangs out with friends while still in costume — in part because the actor, Bryan Batt, had been one of those chorus kids.

“I’d enjoyed the show, and when I’d mention this among any group of sophisticated New Yorkers they’d have a fit,” Rudnick says. “I think everyone loves talking about ‘Cats,’ pro or con, because it’s such a fabulously absurd topic — no matter how hard anyone tries to conduct a serious debate, you’re talking about people wearing tails and whiskers.”

Hence the social media consternation over the film’s computer-generated human-cat hybrids. The reviews are no more positive — “misfire,” “garish CGI experiment,” “outlandishly tacky,” fascinating mess,” “exhilarated and baffled” — and the first weekend’s domestic box office was just $6.5 million, so do not expect a sequel (“2 Cats 2 Purrious”).

Clare Reihill, the literary trustee of the T.S. Eliot estate, says that the poet would have liked how the movie portrays the cats, who in the poems evoked London society types. “How does one re-create that world on film unless you imagine the cats as almost like us yet not quite?” Reihill says in an email. That duality “speaks to the surrealist in Eliot, the poet interested in the inversions of the nocturnal world, in the strange collisions of high- and low-life. I think the movie has some of that magic.”

Plus, Eliot recognized the animal’s mysterious quality — a musical called “Dogs” would not have worked as well. “Dogs are sweet and loyal, but they’re not sneaky. They’re not up to something,” says Katie Hill, who in 2012 conceived the Internet Cat Video Festival. “There’s a personality factor, that aloofness of real cats, that lends itself to more character building.” It might be no coincidence that “Cats” epitomized guilty pleasure in theater, just as cat memes and videos have done the same for the Internet.

But maybe we shouldn’t overthink it. “It was so entertainingly ridiculous — singing and dancing are the last things any real cat would ever be interested in doing,” Rudnick says. “There isn’t that much Fancy Feast in the world.”

This story has been updated.

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