The Oval Office meeting between Richard Nixon and Elvis Presley in December 1970 to this day remains an intensely fascinating moment in pop culture history, shrouded in mystery and packed with potential for grand dramatic license, as there’s no official record of what the men discussed.

There’s an equally large possibility of descending into caricature in telling the story of “Elvis & Nixon,” though.

So the first small miracle achieved by director Liza Johnson and stars Michael Shannon (Elvis) and Kevin Spacey (Nixon) in this feature imagining how that meeting might have gone, is that they studiously avoid single-dimensional portrayals of two of the all-time most famous and oft-imitated Americans.

The picture unpacks the bizarre circumstances surrounding the meeting — Elvis’ conviction, at some point after watching “Dr. Strangelove,” that his country needed him to fight the scourge of drugs; his sudden arrival at the White House to deliver a letter to the president at 6:30 a.m. — with the funky spirit of a B picture from the period, with a brassy score and the frenzied pace of a parody.

Johnson smartly orchestrates the chaotic scramble inside the White House to convince the president that some “damn entertainer” was worth interrupting his nap hour, with aides (Colin Hanks and Evan Peters) not above resorting to involving Nixon’s daughter Julie.

At the same time, Elvis’ plight in D.C. is largely regarded through the eyes of his friend and confidant Jerry Schilling (Alex Pettyfer), who supports the King in his mission to pitch the president on becoming an undercover agent not out of some inexplicable conviction in its merits, but because he’s a friend that’s just what friends do.

And that’s the key to making sense of Shannon’s take on Elvis, defined by a sort of core loneliness shared by a president forever suspicious of everyone. It’s hard to truly know yourself if it’s impossible to have a genuine connection with anyone.

Spacey, a gifted mimic, plays down Nixon’s characteristic hangdog quality in favor of a sort of bemused incredulousness at this pompadoured man in a garish outfit facing him in his office.

The movie is defined by Shannon, though, who finds a way to universalize the experience of being Elvis without totally downplaying the abiding weirdness.

Shannon’s Elvis is a man in search of himself, experiencing the wearying effect of an unending battle between the character he plays for his fans and the reality of the boy from Tupelo, Mississippi, that’s still there, somewhere, underneath all that flare.

It’s in that context that the movie finds a deeper plane of existence and gives logical shape to the illogical nature of Elvis’ desire to step away from the limelight and take on the drug culture: It’s about a man trying to be the man he wants to be and not the one everyone expects.