Here, for the centenary of Ingmar Bergman’s birth, is a rerelease of one of his fiercest, strangest, most sensually brilliant and unclassifiable pictures: Persona, from 1966. This was last revived in British cinemas 14 years ago, and I have in the past been agnostic about what I felt were contrivances and rather atypical attempts to engage with the Godardian spirit of the times.

Revisited now, the movie actually more suggests the Roman Polanski of Knife in the Water and Repulsion. Yet more than that, it forces on the audience its own utter uniqueness. It is stark, spare, endlessly questioning and self-questioning, a movie whose enigmas and challenges multiply, like the heads of Hydra.

It begins with a disturbing montage of unreadable and occasionally sexually explicit images, which are reprised briefly at the moment of psychological crisis later in the film – and which only partly explain them. A delicate boy on a mortuary slab may be a certain male baby boy that a guilt-stricken mother wishes dead. Images of silent-movie histrionics and pantomime absurdity may or may not gesture at the shallowness and falsity of acting, the theatre, the cinema and all representational art. Or they may indicate that these cartoony, melodramatic images can give a lightning-flash of clarity, of fundamental truthfulness and interpretation, that life itself, in all its evasive complexity, may never yield to us.

Liv Ullmann plays Elisabet Vogler, a renowned stage actress who has suffered a psychological breakdown. She relapsed into silence during a production of Electra (she was probably playing the lead, but possibly Clytemnestra in a more stylised production, it is not entirely clear) and is now in a psychiatric facility. Bibi Andersson plays Alma, the nurse assigned to her. The doctor supervising her (Margaretha Krook) has a bold and generous idea – one which she perhaps might not have offered for a less celebrated patient. She will lend Elisabet her handsome summer beach house for the duration of her recovery, and Alma will go and live with her.

It is to be an extraordinary inversion of the “talking cure”. Far from being coaxed out of her silence, Elisabet remains utterly mute, and it is Alma who begins to speak. At first in an artless attempt to get Elisabet to open up, but then she finds it is her own necessary personal catharsis. Alma tells Elisabet about the problems and crises in her life, while Elisabet remains enigmatically – though eloquently – wordless.

Actually, she is not quite wordless. Alma begins to confess that, though now engaged to be married, she has already had her heart broken in an affair with a married man, and also had an unforgettably erotic encounter on a beach while sunbathing naked with a friend. Alma is upset to read a letter from Elisabet to the doctor, which she had carried into town for the purposes of posting, which is rather dismissive and amused about these personal revelations. Alma’s growing love for Elisabet curdles into resentment and rage, and she might also suspect what Bergman surely wants the audience to suspect – Elisabet had left the letter unsealed precisely so that Alma would read it.

Alma is angry, hurt, frustrated, and perhaps even more irreversibly and painfully in love with Elisabet than ever – and more than ever desperate for Elisabet to speak, to communicate, to indicate that she respects and understands what Alma is going through. Or perhaps simply to acknowledge her existence. Their intimacy becomes a kind of duel, yet also a kind of fusion. Are they dreaming each other’s existence? Alma says at first that she respects Elisabet’s silence as an ethical or moral position that has developed from her approach to life as an artist: if she cannot uncover the hidden existence of others, then she will at least withdraw into silence and conceal her own inner existence. Or at any rate, not go through the meaningless charade of appearing to reveal herself through the misleading parade of conversation.

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At another point, Alma muses in a besotted, almost girlish, way about what it must be like to be Elisabet. She can see how a humble person like her could fantasise about inhabiting the great actress’s body, while acknowledging that it is the artist’s prerogative to impersonate an ordinary person like her. And all the time, they grow mysteriously closer. The psychological pressure becomes ever more intolerable, and Elisabet is further challenged by harrowing images from the outside world. She chances upon the famous photo from the Warsaw ghetto in 1943 of a Jewish boy with hands raised, and recoils in horror from TV news footage of the Vietnamese Buddhist priest Thích Quảng Đức, who set himself on fire in a Saigon street. These are, I think, moments in which Godard’s influence, and the 60s zeitgeist, is very plain.

Bergman contrives a colossal eerie closeup with meshes half of Alma’s face with half of Elisabet’s. These two very beautiful women do indeed look similar, and it is that fact which makes the resulting half-and-half face so horrifying, like something from a nightmare. Elsewhere, Bergman repeatedly gives us the signature composition of the two faces: one in profile, one face on to the camera, sometimes overlapping. It is almost a pictorial demonstration of the film’s approach to identity and disclosure. Sometimes the characters are facing us, addressing us, wishing to reveal themselves. At other times, they are in profile: we can see them, but they are looking away, at that moment indifferent or unaware.

Persona is a film to make you shiver with fascination, or incomprehension, or desire.