Some movies have no secrets to keep and are eager to tell you so, very loudly. Especially when summer is near, they come trooping into theaters with sequel numbers trailing their titles, your guarantee of seeing something that will play like an extended cut of its own trailer.

Better films, of course, prefer to hold something in reserve, and may sometimes even seem reluctant to let their secrets go. That’s the case with Jonathan Glazer’s remarkable Under the Skin, which makes you wait in suspense to seize any scrap of information that might explain the strange comings and goings. More than that, though, you ache for the emotions to stop tantalizing you and slip into your grasp.

And then, every once in a while, you come across a film like Pawel Pawlikowski’s Ida that never gives up its secrets at all. People reveal their pasts; evidence emerges of terrible crimes; and yet the movie seems to stand quietly apart from its own events, never allowing you to see fully into the characters or to know for certain what meanings their lives might hold for you.

You have to work a little even to figure out the period in Ida. You know at once that the movie is set in an era appropriate to an old-fashioned, squarish screen format and black-and-white cinematography (or almost black-and-white, I should say, given the ghosts of color that infrequently seep in). From some remarks that are dropped in a brief courtroom scene, you can confirm that the film’s Poland is still communist; and from the tailored cut of people’s clothing and the humpbacked contours of their cars, you can estimate when the story takes place within a range of fifteen or twenty years. (Styles weren’t exactly churning in Poland between Ashes and Diamonds and Camera Buff.) But it’s only when you learn that the title character was born during World War II and you surmise that she would now be 18—the right age for this convent-raised novice to take her vows as a nun—that you can at last narrow the date to around 1962.

But by saying all this, I’ve already violated the mood of undecidability that’s so important to Pawlikowski. Contrary to what I’ve just said, there really is no title character in Ida—or rather, she does exist, but she goes by the name of Anna.

Cat-faced and cleft-chinned, with dark eyes that seem a little wider because of the lowering of the novice’s headcloth across her forehead, Anna first appears in close-up on a screen that’s depthless and almost black, her smooth skin glowing within one of the shafts of almost palpable white light that characterize the film—a Carl Dreyer spiritual light, you might say. She’s isolated within the shot, except for the figure she’s studying with rapt attention: a statue of Jesus. Over the next minutes, as a clipped succession of static views presents life in the convent, you come to understand that Anna is both alone and not alone, joined to a community but attached only to God. Pawlikowski makes every material fact of her life feel emphatically present—in the refectory, for instance, where the scraping of spoons against bowls becomes an all-encompassing clatter—and yet lets a sense of absence hover everywhere. Throughout the film, he composes images that are dominated by the empty air occupying a gray-walled room, or an immensity of winter clouds looming above a landscape. Meanwhile, he pushes his people to the margins: relegating them to a narrow band that stretches across the bottom of the frame, perhaps, or a thin strip of land in front of a low horizon.

This aura of paradox takes hold even before the mother superior orders Anna to go to Lodz in preparation for taking her vows, so she can meet her only living relative: the aunt who has always refused to visit her in the convent. Meekly, dutifully, with great simplicity, Anna takes a bus into the city and ventures up to the apartment of Wanda Gruz, who had summoned her and yet seems barely willing to open the door. Not that anything about her place would welcome a young woman in a habit. A mambo is playing on the radio; a man, presumably not Wanda’s husband, is slowly dressing in the bedroom. With a cigarette in one hand, a glass of vodka in the other and her slender frame wrapped in a bathrobe, the hard-faced, 40-ish Wanda leads Anna into the kitchen for as much of an interview as she appears willing to grant. “So,” she finally says with something like revulsion, as she takes the film’s paradoxes to their next level, “you’re a Jewish nun.”

The remark doesn’t take long to explain: Anna, born Ida Lebenstein in a rural village, had been conveyed to the convent as an infant after her immediate family met the fate of most Jews. Once this bare fact has been disclosed, though, its implications require an entire road trip to play out, and a tense, complex, sometimes angry coming to terms between the two women—a negotiation that begins, none too promisingly, with Anna’s declaration that she will visit her parents’ graves, and Wanda’s retort: “They have no graves.”

Innocence, meet Experience. For much of the film, the contest between the two isn’t even close, despite the handicap that Experience gives herself with drink and bitterness; and so the further the characters venture into the drizzly Polish countryside, and the deeper they dig into the past, the more you think that the movie could more appropriately have been called Wanda. Played by Agata Kulesza with superb sarcasm, Wanda is literally the driver of the story (though she does manage at one point to deposit her car in a ditch) and is figuratively its instigator, interrogator and provocateur. By profession, she is a high-ranking judge, whose handsome features—now carved out by long despair and too little interest in solid food—will sometimes draw into a scowl that would freeze any defendant. “Red Wanda,” they used to call her, she explains in a confessional moment with Anna. In the aftermath of the war, she sent many people to their deaths. (“What people?” Anna wants to know. “Enemies of the state,” comes the dry, self-disgusted reply.) Now, with her prosecutorial instincts reawakened to a pitch of long-deferred fury, Wanda drags Anna along on an investigation whose probable outcome she seems to know in advance, but whose details she believes will be educational for a Jewish nun. While she’s at it, Wanda also proposes that atheism, slinky clothes, dancing and a little sex might prove instructive.

What is Anna to Wanda? An object of scorn, a last vestige of love, a burden of guilt, a vehicle of retribution, a woman and a Jew to be brought back into the world. And what does Anna think about this torrent of uses and identities that Wanda is imposing on her? That’s the essential secret Ida refuses to give up. Although the film has material facts to uncover, of a sort that will not be unfamiliar to anyone who knows the first thing about the Holocaust, the answer to the real question, which concerns Anna’s soul, remains shrouded in mystery. You see Anna deal with certain consequences of her time with Wanda; you watch her make certain decisions about her vows. But as played by the young Agata Trzebuchowska, a nonprofessional recruited for her first film role, Anna is a pond of very few ripples and unknown depth. By the end of the film, you know she has changed inside, but the import of that change remains entirely her business—which is the aspect of Ida I most admire, and also the one that makes me most uneasy.

Let’s say that Pawlikowski has either respected the imponderability of Anna’s situation or, less honestly, has left the interpretation of her choice to the prejudices of each viewer. He’s certainly made it possible for a wide range of ticket buyers to justify their own feelings. How much blame adheres to salt-of-the-earth Poles for the mass murder of Jews? How much pity should we muster for the perpetrators of postwar show trials? Is a life of faith the true hope of a hopeless world, or a fallback position? Is Catholic devotion more pure than Jewish self-perpetuation? Imagining the reception of the film in Poland, I can think of an even more inflammatory question: How Jewish were the communists, anyway? Pawlikowski (with his co-writer, Rebecca Lenkiewicz) implicitly raises all of these issues and then leaves them unresolved behind Anna’s silence and the smooth, unreadable screen of her face.

Whether you give the benefit of the doubt to Ida will therefore depend on whether you think of doubt as a benefit. I believe that moviegoers who do not rush to fill in the blanks with their own preconceptions—who recognize that the film operates primarily on the level of ambiguous personal interactions and cloudy motives—will feel that the saving grace of Ida is precisely its reticence before the unknowable. It’s the quality that atones for the plot’s occasional straying into melodrama. It’s the element of self-restraint that redeems Pawlikowski from having indulged in so exquisite a style. With each new shot and each deliberately chosen note of music layered into the soundtrack, Ida demands to be acknowledged as an art object—which is not, perhaps, the most worthy ambition for a film that involves digging up corpses. And yet, in defense of Pawlikowski, I note that Ida’s doubts extend even to itself.

A perfect work of art, the “Jupiter” Symphony, runs throughout the film, repeatedly used by Wanda to plug the void that booze and casual sex won’t fill. It works for her, apparently, but only up to a point. In the world of Ida, even Mozart ultimately fails—and so, by implication, must Pawlikowski.

As for the potential helpfulness of truth-telling, retribution and God, we just don’t know the odds, according to Ida. Seems likely enough to me.

* * *

Dressed in a faux-fur jacket, tousled brunette wig and lipstick the color of fresh blood, Scarlett Johansson drives a white van through Glasgow and its outskirts in the genuinely creepy Under the Skin, scanning for men who are on their own and offering them rides. Something about this behavior is seriously wrong, as you may guess from its round-the-clock nature, the nerve-twisting violin motif that accompanies it, and the way the shotgun-side passengers keep disappearing.

You would expect that a young woman who picks up strange men is putting herself in danger—and by the end of the film, you will learn that Johansson is indeed vulnerable. For most of Under the Skin, though, some conspiracy is gaming the ordinary pattern of sexual attraction and male-female predation for purposes that director Jonathan Glazer leaves mostly to your imagination. If you’re looking for languorous, sensuous menace, he’ll give you all you’d like, with the help of his co-writer, Walter Campbell, and the source novel by Michel Faber. If you want information, though, Glazer will gradually, almost grudgingly, supply little more than some views of an uncanny ritual, which suggests that whatever Johansson might be, she (or it) isn’t human.

I suppose the word “alien” has to come up at some point. If so, we’re talking about extraterrestrial beings that have traveled here on rings of light (the only method of transport Johansson seems to use, other than the van and an occasional bus) and that inexplicably need a supply of human flesh, though not necessarily for food. Although an experimental nibble of chocolate cake turns out badly for Johansson, that doesn’t mean her victims are being flushed into a vat of nutrients. Maybe they’re used as fuel, medicine, perfume or recreational drugs.

All we know for sure is that past a certain point, Johansson relents in her collecting. She has begun to stare curiously at her own reflection. Perhaps she has developed some fellow feeling for her prey. If so, I can say that Under the Skin also puts the viewer in touch with what it means to be human, and in startling ways. There are images in this film unlike any I’ve seen before: veils of light made from the movements of thousands of people, lacquered traps that can engulf men mesmerized by desire, the human form emptied into a twisted, drifting husk. These were all new to me; and so, too, were Glazer’s evocations of our fear, brutality, frailty and kindness.