Unlike The Crying Game, where the sex of a character is a major twist about halfway through, the genders of the characters in Albert Nobbs is not much in doubt: Glenn Close is a big star with above-the-title billing — her butched-up face is the ad campaign. And yet there is just as much mystery here, albeit of a different kind. This is a story of identity that’s almost impenetrable.

Albert (Close) is a gentlemanly servant at a high-end boutique hotel in Ireland. Everyone admires Albert: The women appreciate his respectful demeanor, his male co-workers his work ethic, the boss, Mrs. Baker (Pauline Collins), his reliability. But no one really knows Albert, who lives in a small room in the attic and squirrels away his money and dreams of something else.

But really, Albert doesn’t even know himself. He has been living as a man for decades — who knows how long? — and cannot even remember a time when he (or she) was not Albert. He has become so repressed, he almost doesn’t have a personality anymore.

But life intrudes. Albert finds himself attracted to a young chambermaid (Mia Wasikowska) for whom he yearns; he’s also being hit on — subtly — by another butler at the hotel, who senses in Albert a closeted nature… just not the right one. But the real upsetting of the apple cart comes in the arrival of handyman Mr. Page (Janet McTeer), a tall, laser-eyed fellow who seems to look through Albert. When Mr. Page gets assigned to share a room with Albert, his life is about to unravel.

But Mr. Page has a secret of his own: He is also a woman living as a man. And for the first time perhaps in his adulthood, Albert has a confidante.

Director Rodrigo Garcia takes an unfussy, humanistic approach to the material. He seems happy to allow you to wonder about Albert’s past, unconcerned about tying things up neatly. He teases us with lots of little back stories: Two bon vivant couples (seemingly male-female pairings) reveal themselves as something else; Mr. Page’s private life is explored but not carefully explained. It’s a tribute to him — and to Close, in her best film role in two decades —  that we care without knowing more.

Close carries the film, even in its slower parts and those that begin to devolve into Masterpiece Theater cliche. It’s a tightly-wound performance, in contrast to McTeer’s bold, open one. They make excellent foils, tenderly and realistically showing a new side to gay life, and reminding us that closets aren’t the only places people have to hide.

Now playing at the Angelika Film Centers/Mockingbird Station and Plano.