About 20 minutes into the second episode of the new season of “The Crown,” President Lyndon B. Johnson, a head of state slightly more profane than the queen of England, unzips his trousers at a urinal.

As he audibly dribbles through the process of relieving himself, Johnson’s chief of staff updates him about some tension with the British government.

Johnson, on the show and in real life, is peeved that Prime Minister Harold Wilson isn’t supporting his Vietnam policy. Wilson, on the show and in real life, is worried his country is about to go broke. Both sides need a way to make nice. Johnson’s chief of staff has found one.

After Johnson zips up, his aide shows him the front page of a newspaper. There’s a huge picture of a beautiful, smiling woman.

“Who in God’s name is that?” Johnson says.

It’s Princess Margaret, sister of Queen Elizabeth II. She’s on tour with her husband in the United States — eating lavishly, drinking mightily, burnishing her image as a carefree royal bad girl.

The opportunity, as Johnson’s chief of staff sees it, is obvious: Invite her to the White House for dinner.

Party time!

Fact: Johnson not knowing who Princess Margaret was — that’s total fiddlesticks. The two had met and chatted a few years earlier in Jamaica. Four Pinocchios.

Fact: Princess Margaret and her husband did attend a state dinner at the White House, though it was a long planned evening and not tied to any dispute over Vietnam or England’s finances. According to historians and Princess Margaret biographers, it was a wild night, though not nearly as entertaining as “The Crown” depicts.

For instance, there is no record of Princess Margaret insulting the memory of John F. Kennedy, for whom Johnson served as vice president before his assassination. The fictional moment does allow the show’s writers to put these words in Princess Margaret’s mouth: “You spent three years as vice president. I’ve spent my whole life as vice queen.”

The poor thing.

There is also no record or recollection by anyone present of a limerick contest between the princess and the president — a match of wits and rhyme employing a woman’s breasts and a man’s “Johnson.”

“It’s certainly not something I’d ever heard of,” Christopher Warwick, the author of a Princess Margaret biography, recently told Vanity Fair.

Still, much fun was had. Kirk Douglas was there. Nelson Rockefeller, too. And Henry Ford. Johnson’s official White House diary lists the menu:

“Apparently there was dancing,” the on-screen queen tells Prince Philip, recounting how she was upstaged again by her sister.

Fact: There was dancing.

“The President danced quite a bit during the evening, and was not sitting in his chair very much,” the White House diary says. “He had a good time, he looked well, and danced with almost every woman there.”

Princess Margaret and the president foxtrotted to “Everything’s Coming Up Roses,” according to the Newcastle Evening Chronicle, which described their foxtrotting as “enthusiastic.”

But did they kiss on the lips, as they did on screen? Not according to any biography of Johnson or Princess Margaret reviewed by this fact-checker.

Still, one can dream.

Anyway, the night capped off a truly memorable trip for Princess Margaret and her entourage — so memorable that she was “barred by British diplomats from making an official visit to America in the early 1970s because of the wild behaviour of her entourage during an earlier visit,” according to the Telegraph.

As for the president, he probably slept in the next day. The evening, according to the White House diary, stretched well into the morning:

Read more Retropolis: