A cerebral sci-fi time-travel puzzler that proves consistently engrossing and, for all its contrivance and complication, satisfying, “Predestination” represents a considerable success for its fraternal writing-directing team of Peter and Michael Spierig and star Ethan Hawke, whose willingness to undertake unusual projects like his films with them and Richard Linklater have made him one of the more interesting American actors around. But it’s a special triumph for Sarah Snook, a little-known Australian actress whose turn should do for her career what Edward Norton’s curiously similar one in “Primal Fear” did for his.

The Spierigs and Hawke, of course, worked together before in “Daybreakers,” a 2010 vampire movie that, despite some twists, didn’t transcend the limitations of what was already a tired genre. While time-travel isn’t exactly a new concept, and even the idea of a time-travelling cop has been around before (it wasn’t even fresh when Jean-Claude Van Damme used it twenty years ago), the Spierigs, working from a 1960 short story titled “ –All You Zombies— “ by Robert A. Heinlein, have been able not only to construct a labyrinthine tale that leads (or rather jumps) from one surprise to another but to weave them all together into what proves a goofily plausible whole. And they direct their script in a style that recalls early Cronenberg in its mixture of matter-of-factness in the face of absurdity with a sort of hallucinatory queasiness. This is a film that doesn’t pound you with sledgehammer point-making, but lures you deeper and deeper into its ever-weirder web, pulling the rug out from under you repeatedly but leaving you happy to go along with the jolt and move on to the next level.

Revealing too much about the plot would be unpardonable, but it ruins nothing to say that much of it revolves around a terrorist known as the “Fizzle Bomber” blowing up sites in New York City during the 1970s. The first scene shows a man in a trenchcoat trying to stop the villain from setting his latest device in motion, but suffering terrible effects from the explosion when he fails. A shadowy figure, however, assists him to reach a device that transports him to a hospital where his face is reconstructed into that of Hawke, and he’s identified as a Temporal Agent who travels through time retrieving clues about the terrorist that might allow him to be caught. Before long he’s healed and is back on the job, working as a bartender in a NYC dive.

There he encounters a strange-looking, bad-tempered customer (Snook) who identifies himself as the writer of confessional stories that travel under the byline of “the Unmarried Mother.” During a long conversation, the fellow pours out his life story—which begins with his childhood not only as an orphan but as a girl. Told in elaborate flashbacks, the tale involves seduction, pregnancy, a kidnapping and gender transformation—as well as a stint as a recruit in a company looking for exceptionally talented women to serve the needs of astronauts in space, a prospect presented to her by a quietly authoritative character called Mr. Robertson (Noah Taylor). After hearing the bizarre recounting, The Agent offers Mother a chance to confront the man who ruined his life—and even to kill him.

Up to this point “Predestination” is odd but fascinating; from it, the picture becomes odder and even more fascinating. It wouldn’t be fair to disclose the direction it takes, or the shifts that follow; and while one may have doubts about how it winds up, precisely what those doubts are should be kept to oneself, at least until you’re debating them with somebody else who’s also seen the movie. Suffice it to say that the rules you’ve probably been told are essential to the very notion of time travel don’t necessarily apply here. And though this movie might change those rules, it plays pretty fair by the new set.

Whatever reservations one might harbor about how the narrative is worked out, in any event, it’s well-nigh impossible to have any about the amazing quality of Snook’s performance, which not only rescues a character that might have become little more than a joke but gives it emotional heft. Hawke supports her with a turn that’s mostly straightforward but intense when it needs to be, while Taylor coolly embodies the ultra-capable company man who recognizes potential—and danger—when he sees it. The rest of the cast handle their roles capably. And the behind-the-camera crew do very effective work on what was clearly a modest budget, from Matthew Putland’s production design and Janie Parker’s art direction to the sets designed by James Parker and decorated by Vanessa Cerne and Wendy Cork’s costumes. Ben Nott contributes elegantly unfussy camerawork and Matt Villa subtly paced editing, while Peter Spireig’s score is nicely understated. Nor should one overlook the outstanding contribution of makeup designer Steve Boyle.

“Predestination” may not explore very successfully the deeper issues about the human condition that it wants to touches on. But it works astonishingly well as an extended “Twilight Zone” tale told with consummate skill.