It’s just a tiny slice of pop culture trivia, meaningless in the greater historical sagas of the lives of The President and The King.

But you can’t look at that photo of Elvis Presley meeting Richard Nixon in The Oval Office and not laugh. The “How” and “Why” of this meeting, and the “What on Earth did they talk about?” would make a helluva story.

And so it does in the deft and delightful “Elvis & Nixon,” a short, quick and clever recreation of the how that came to pass and an imagined version of the conversation that could have taken place.

This lightweight, laugh-out-loud farce allows two of the greatest actors working in movies to try on the guises of two cultural icons, and find the humor in their self-delusions.

Michael Shannon isn’t as pretty as the real Presley. He doesn’t bother much with his Deep South drawl. But he plays the delusions, the ennui, and the last vestiges of “Cool Elvis.” This Elvis is a country boy who never grew up, with the naivete of a guy who figured he could leave a letter at the White House gate and get an audience with The President, then and there.

Because…ELVIS.

As Christmas of 1970 rolled around, Elvis was looking at the sorry state of America and deciding he was just the fellow to set things right.He will get Nixon or J. Edgar Hoover or somebody to appoint him “Federal Agent at Large.” Hell, he’s acted in 31 movies. He’d be a cinch at undercover work.

And “I can supply all my own firearms.”

Since we meet him as he’s silencing three TVs at Graceland with his pistol, we know that’s true.

Elvis, who has been deputized by every sheriff and police chief within reach of Memphis, wants a Federal badge. And his pal Jerry Schilling (Alex Pettyfer) is just the guy to get it for him.

Jerry used to be in the “Memphis Mafia,” the Elvis entourage. He’s escaped to Hollywood, working in film editing at Paramount. He has a girlfriend and a life. But when “E” calls, Jerry delivers. Besides, Elvis has already thought this through. He flies West to fetch Jerry, writes Nixon a letter and all they have to do is show up in “Our Nation’s Capital” and deliver it.

Shannon gets to play a chillingly self-aware Elvis. There’s no suggestion of the drugs that he became famous for, post mortem. This guy knows the world sees him as “a thing, like a bottle of Coke.” He is used to his status. There is only one “King.” And he’s used to getting what he wants, no matter how delusional that seems.

Spacey plays Nixon as something more than the profane parody he has become in history. Stoop shouldered, jacket buttoned and bunched-up, even at his desk, he’s salty, all-business, and not wasting a minute of his time on this crazy pitch that his aide Egil Krogh (Colin Hanks at his most straight-laced and amusing) has come up with. Elvis made the first move. He can help with “the youth vote” and so on. Why not?

Director Liza Johnson and the screenwriters (including actor-turned-writer Cary Elwes) package this as a mirrored tale of two men equally out of touch. Elvis was way past “the youth vote,” and King or no King, there’s no way Elvis Presley should be able to talk himself into seeing The Leader of the Free World and scoring a badge from the Feds, just on a whim. Who does he think he is? Bono?

Much of the film is the comic build-up to the meeting, the horse-trading involved when a King and President meet.

But the biggest laughs come from the women, mostly in the background of this story as they were in American society. Airline ticket agent to “Federal Bureau of Narcotics and Dangerous Drugs” office staff, White House aides to Nixon daughter, they’re all agog at the merest hint that Elvis hasn’t left the Building. They blush, gasp and giggle and we giggle.

It’s mostly fantasy, at least in several important ways. No drugs were in the system of the rock star who wanted to bust drug users with a Federal badge? Really? Look at the real photo. He was on…something.

But Shannon and Spacey have a blast giving these over-familiar characters new twists.Their delight spills off the screen in their big scene together — it is squirm-inducing, in all the funniest ways. And that delight is shared by all in this whimsical riff on a piece of utterly inconsequential American history, a Trivial Pursuit question at its most trivial.

MPAA Rating:R for some language

Cast: Michael Shannon, Kevin Spacey, Alex Pettyfer, Colin Hanks, Johnny Knoxville, Ashley Benson, Sky Ferreira
Credits: Directed by Liza Johnson, script by Joey Sagal, Hanala Sagal and Cary Elwes. An Amazon Studios/Bleecker Street release.

Running time: 1:26