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Dear God, what have they done to Joseph Gordon-Levitt’s face? In Rian Johnson’s time-travel action drama, “Looper,” the star has been prosthetically altered with a fresh set of eyebrows, a snubbed nose, green contact lenses, and what appear to be new lips. He’s supposed to look like the young Bruce Willis. What he resembles, mostly, is mid-period Devo.

The imposture is hardly convincing, but “Looper” is fast enough, weird enough, and just about smart enough to make you forget about that. The movie wants to mess with your head, depositing us in a decrepit, overcrowded 2044 and then sprinkling on time-loop variations like jimmies. Gordon-Levitt plays Joe, a coolly feral young hit man who works for a crime syndicate that exists 30 years in his future, a period when time travel has been both invented and outlawed. These unseen gangsters put a sack over their victims’ heads and send them back three decades, where Joe or one of his colleagues is waiting with a shotgun.

There are plenty of better ways to dispose of a body, and, if you give it a second’s thought, you’ll come up with them. Still, that’s the setup, and “Looper” pedals furiously and largely successfully to keep our disbelief from crashing to the ground.

Joe has had a sweet life with the syndicate — its point man in 2044 is a weary Jeff Daniels, playing a character who has literally seen it all before — and the hero’s fine with the fact that he has an expiration date. When the future gangsters are done with one of their assassins, they send him back in time as a “looper” to be killed by his younger self, who then gets a hefty payment in gold bars and a long time to spend it well and unwisely. The movie’s plot kicks into overdrive when a hit comes down the time tunnel to Joe and it turns out to be him. Or him plus 30 years. Old Joe (Willis). Confused yet?

If not, you will be when the film immediately rewinds and shows us the events as Old Joe sees them — memories of the intervening years, an unexpected lover (Summer Qing), and then the sudden rush to stay alive in his own past. It seems there’s someone who’s an adult in the future who needs killing as a child here and now. In a way, “Looper” is “The Terminator” seen from the cyborg’s point of view, but it doesn’t get too hung up on the metaphysics. “I don’t want to talk about time travel,” Old Joe snarls in one scene. “We’ll be here all day making diagrams with straws.”

Johnson is a prodigious talent who keeps coming close to a breakthrough. His first feature was the 2005 high school noir “Brick” — Gordon-Levitt as Bogart in blue jeans. His second, “The Brothers Bloom” (2008), was a con-man romantic comedy so self-consciously clever you wanted to smack it. “Looper” is his bid for the big time, a conceptually audacious brain-twister with smarts, style, “Matrix”-wannabe cool, and action sequences that click into place like a Rubik’s Cube. It’s something to see yet slightly less than the sum of its parts. Johnson has a great movie in him, and he keeps getting closer, but he’s not there yet.

The two stars are fun to watch, and you can tell they’re energized by the challenge of playing the same character at different points on the grid. Gordon-Levitt effectively muzzles his affability and becomes a colder, harder figure — an action hero stamped out of industrial-strength plastic. Willis’s performance isn’t a match for the pug nobility of “Pulp Fiction” or “Twelve Monkeys” (another time warp tango), and you can see his limitations as an actor when Old Joe is called on to express emotional agony. That said, all this film’s multilevel daring comes to a head when the two Joes face off over a diner table; it’s the mirror scene from “Duck Soup” for an age of personality crisis.

Traces of the director’s love for old detective movies linger. As a quavering loser of a fellow hit man, Paul Dano summons the ghost of Peter Lorre, while Noah Segan as a dumb but pretty enforcer is straight out of the Elisha Cook Jr. playbook. But at least Johnson is going for something fresh. He’ll throw out an image like the letters that appear scarred into Old Joe’s arm and he’ll force you to do the math — that this is the young Joe’s way of getting a message to his older doppelganger. “Looper” crackles with the wit of a talented filmmaker trying to out-think himself.

You wonder how long Johnson can keep up the pace, and, truth be told, he doesn’t. After a first half so hectic and confident you can feel your synapses being redirected into new pathways, the movie settles down into a dystopian spaghetti western. Young Joe comes to ground at a farm outside the city, knowing that his quarry/nemesis/self will arrive in due time. The farm is owned by a trigger-happy but soulful single mom named Sara (Emily Blunt, nicely distressed) and her young son Cid (Pierce Gagnon). The kid has an old face: Whatever environmental or industrial cataclysm has created this ruined world, we see the damage most clearly in Cid’s eyes.

Something like a love story ensues, and while “Looper” briefly ponders whether young Joe’s new feelings for Sara will erase Old Joe’s memories of his wife, the butterfly-effect games largely slow to a halt. From a series of baroque curlicues, the movie straightens into a line, and the heady inventiveness of the early scenes dissipates. The final twist is both logical — inasmuch as anything in this movie is logical — and disappointingly abrupt, as though another shoe were hanging up there refusing to drop. With “Looper,” Johnson proves he can finesse the most complicated notions and visual setups his mind can imagine. It’s the simple things that still elude him.