What elevates Sisters above a standard Hitchcock rip-off, and makes it authentically De Palma, is its typically unsubtle and scathing social critique. Latching on to the disillusionment of late-1960s America amid the broadcasting of the Vietnam War, he makes his concerns about morbid fascination apparent. By framing the game show scene as a television studio set, De Palma positions us as the live audience, making it impossible to ignore his on-the-nose satire.

As evidenced in this scene, the act of looking is central to the plot of Sisters and is unavoidable as a by-product of its Psycho-meet-Rear Window narrative. Crucially, budding investigative journalist Grace Collier (Jennifer Salt) is introduced as the lead character through a voyeuristic episode: the viewing of Woode’s murder at the hands of psychologically unhinged Dominique, Danielle’s since-separated Siamese twin and ulterior personality.

After a frenzy of blood and Bernard Herrmann’s screeching score, Collier’s identity as Danielle’s distant neighbour, and witness, is revealed via a tantalising zoom out. De Palma uses his notorious split-screening to show this ordeal unfolding from both apartments simultaneously, suggesting that no one is immune to being spied on in this twisting, slasher escapade.

This begins the reporter’s story of female crisis as distrust haunts her at every turn: whether it’s the police, her own mother, or a private investigator she hires. De Palma makes this disbelief as frustrating to watch as possible, to both point at a repressive patriarchy and indicate the dangers of her invasive, suspense-fuelled investigation.

He achieves this by frequently positioning the audience as voyeurs of the story rather than being immersed in one character’s perspective: effectively punishing them for peeping too. One excruciating split screen sequence shows Grace’s fracas with the police side-by-side with Danielle’s ex-husband Emil (regular De Palma collaborator William Finley) hurriedly covering up the murder and hiding the body in a sofa bed.

Collier’s persistence heightens upon viewing a Breton Twins exposé, reminiscent of the exploitative real-life documentaries that fed perverse curiosity in the sixties. Her own morbid fascination leads her into captivity under Emil and she is forced to witness the journey of the twins whilst sedated in a harrowing dream sequence.

This is where De Palma really flexes his directorial muscle, physically launching Collier into the perspective of Dominique through her own pupil, which transitions to become a peep hole into the twin’s nightmarish history. Switching from 35mm to 16mm, colour also ceases to exist as she witnesses life on the other side of the lens, including the invasive recording of the documentary and the neurotic behaviour of the general public.

Such an explosive crescendo rounds up what De Palma was saying all along: our voyeuristic tendencies are unhealthy, perverse, and ultimately dangerous. The final shot and the last act of looking is one of obsession and zero resolution. Through binoculars, the private investigator observes the sofa bed intrinsic to the murder. No one is coming for it and the case is dead in the water. Still, he watches on.

Sisters opens with a double-bluff. First, as the credits roll to the instantly recognisable dramatic strains of a Bernard Herrmann score, we see up-close stills (courtesy of Lennart Nilsson) of a foetus developing in utero, before it is revealed that there is a second foetus hidden behind the first. Next, in the opening scene, ad man Phillip Woode (Lisle Wilson) is shown in a changing room watching as a blind woman enters and starts undressing – only for it to be revealed that this is a sting operation, with Phillip’s moves and moral choices being filmed by hidden cameras for a television show called Peeping Toms. By turning away, Phillip passes the test of chivalry, and is rewarded with dinner for two at Manhattan’s nightspot. The ‘blind’ woman – in fact, a perfectly sighted Quebecoise model called Danielle (Margot Kidder) – puts herself forward to join Phillip at the club, and then invites him back to her apartment on Staten Island. The following morning, Phillip is murdered there, stabbed with a knife from the cutlery set that had been Danielle’s prize on the show.

These two openings, one short and one much longer, introduce what will prove key themes in Brian De Palma’s first thriller: twins (and other doubles), and voyeurism. It turns out that sweet Danielle is a Siamese twin, surgically separated from her more disturbed sister Dominique (also Kidder) as an adult, and still bearing scars (both physical and psychological) from that traumatic rupture. Linked by their otherness – he is African-American, she is French-Canadian – Phillip and Danielle are brought together by a television show devoted to wandering eyes, and as they spend the night and morning together, they are still being observed – by Danielle’s ex-husband, Emil Breton (William Finley), who possessively stalks the model, by Dominique, who lingers jealously in the next room, and by budding journalist Grace Collier (Jennifer Salt), who lives in an apartment opposite and is partial eyewitness to Phillip’s murder. Even the newspaper for which Collier currently works is called the Staten Island Panorama – that last word signifying the full view of the film’s plotting which, in the end, the audience can see even if the surviving characters remain blind.

It is no coincidence that Herrmann was hired to provide the score. For Sisters, like so many of De Palma’s subsequent films, pays homage to, even makes pastiche of, the thrillers of Alfred Hitchcock. Grace snooping from her apartment replays Rear Window (1954), while the killing and cover-up that unfold in Danielle’s apartment come with echoes of Rope (1948) and a gender-reversed Psycho (1960). If these allusions represent one kind of doubling, there are certainly others, from De Palma’s expert use of split screens to divide the film’s narrative perspective in two to the convergence of Danielle’s and Grace’s storylines until they become hypnotically conjoined. Sisters is a film of two halves, with Phillip’s murder and disappearance followed fast by Grace’s amateur sleuthing – in which she is paired with professional detective Joseph Larch (Charles Durning). Their two-pronged investigation will lead to some crazy places, and indeed to a place for the crazy, where one man’s gaslighting ways are made to coalesce with a psychiatrist’s clinical use of mesmerism, and two very different women will find themselves trapped in the same Shock Corridor.

Both Grace and Danielle are trapped. Grace may have high ambitions to pursue her career as an investigative reporter independent of any men, but voices around her, like the urgings of her mother (Mary Davenport) that she give up her “little job” and get married, keep driving her to conform to the prevailing, male-oriented system, while the policemen and Detective Larch broadly patronise and ignore her. In a different way, Danielle is even more trapped, still mentally conjoined to her now-separate sister as though she were a phantom limb, and caught in her ex’s abusive web of control. The two murders in the film are both presented as acts of revenge – one tragically misdirected – against patriarchy itself. Indeed, both begin with castrating slashes at the male groin, the seat of phallocentric power. Yet, by the end of the film, Grace has thoroughly internalised a male voice that undermines her own best interests – and prevents her from ever being able to publish her big journalistic scoop. For this is a film where sisters, and the sisterhood, are shown to fail in their bid for success or freedom, all thanks to a domineering, manipulative Svengali who gets into their heads and brainwashes them to meekness, madness and murder.

“There’s nothing simple about any of this,” complains perplexed police detective Kelly (Dolph Sweet) near the close of Sisters, unable to comprehend how or why the once determined and dogged Grace is now “just not quite herself” and insists that no crime has even been committed. Perhaps the reason is that the real criminal here, patriarchy, is able, not unlike Danielle’s twin, to continue exerting a malign influence long after it has been cut out of the scene. In any case, De Palma’s brassy, bonkers film will have you seeing double.