The “get,” in the TV business, is that one huge interview every news or talk show wants because of its cultural significance and built-in buzz.

Talk show host Dick Cavett got that “get” with Muhammad Ali … 14 times over several decades.

Their years-long small-screen friendship, which crossed over into their personal lives, is recounted in “Ali & Cavett: The Tale of the Tapes,” a new documentary premiering Tuesday (9 p.m.) on HBO.

“I remember we got into sync from the beginning,” Cavett, 83, tells The Post. “He had an instinct for what I was going to say before I said it. He was just remarkable in that way.

“He really did have a brilliant sense of humor and brilliant show-biz instincts,” he says. “He had all he needed to be a star in the entertainment world — his timing, his selection of what worked to make it just right and when to shut up.

“He was a born showman.”

That’s all on display, and much more, in the clip-heavy “Ali & Cavett,” which includes new interviews with Cavett and others, including sportswriter/HBO commentator Larry Merchant and author Ilyasah Shabazz. It traces the arc of the Ali-Cavett TV interactions dating back to the 1960s and continuing over the years on various iterations of “The Dick Cavett Show” spanning the boxer’s transition from Cassius Clay to heavyweight champ Muhammad Ali to his sad professional decline, including his 1973 loss to Ken Norton, who broke Ali’s jaw. “I only got calls from Dick Cavett,” Ali, his jaw wired shut, says in a visit to the show after the Norton fight. “You are really my main man because this is the only show that invites me right after I get whipped.”

Their televised discussions covered everything that embodied Ali’s fame and public profile — including his three-year suspension from boxing for refusing to be drafted after converting to Islam, the country’s ongoing racial tensions and Ali’s triumphant return to the ring (and bouts with Joe Frazier, George Foreman, Ken Norton et al.) as he became the sport’s only three-time heavyweight champion. There are clips of Cavett’s 1974 visit to Ali’s training camp in Deer Lake, Pa. showing the men clowning around. (Ali beat Joe Frazier in their second fight shortly thereafter.)

“It would be hard to know what he wouldn’t talk about at any time,” says Cavett. “First of all, I had to adjust to how smart he was. He had all of that and almost more than he could handle, and he always knew what to say, what tone to take, when he was on too long and how to manipulate the other person, as he did sometimes a little mercilessly with Joe Frazier.”

One of the documentary’s memorable clips shows Cavett interviewing Ali and Frazier on “The Dick Cavett Show” when Ali says to Frazier, “Let’s be friends for one minute just to get him” before they grab the startled Cavett (one arm each) and lift him out of his chair. “They elevated me so suddenly [that] I saw the floor going away and I was up among the [studio] lights,” Cavett says. “I was thinking, ‘I hope these guys have a good grip.’ That was a wonderful adventure.”

Cavett also shared some personal offscreen memories of Ali with The Post.

“We were in the Caribbean once at the same time,” he says. “He was on a nearby island and I invited him over to lunch to where I was on Virgin Gorda. He stepped out of a small airplane and there was a young guy there, a native of the island, who ran up to him and threw a couple of punches in the air and Ali threw them back. It was like seeing God for some people. We drove around the island and there was a kid standing by the road and Ali said, ‘Stop the car,’ and he rolled down the window and stuck his head out at this kid, who nearly died.”

Another time, Ali spent the night at Cavett’s house in Montauk. “In the morning he came down the stairs and I can never forget it — he was like a giant in a fairytale, this gorgeous three-time heavyweight champion. I made him breakfast — pancakes, bacon, eggs, whatever I figured boxers ate — and as I turned away to fix something on the stove he had eaten everything.

“He said, ‘Dick, was that supposed to be your breakfast?’ It was wicked of him, but he was having fun.”