Sisters (simply due to its being an earlier example) may well be said to have inspired Basket Case or even Twins... But did it, really?
The 'evil twin' seems like a often used trope, but it seems more like a cliche extracted from TV or soap operas rather than something that is commonly used in Horror cinema.
In this case we have a stylish De Palma psycho, suspense, thriller which borrows openly from Rear Window, Psycho & Rope and a score by Bernard Herrmann (surprise!) but whose story in not remotely close to any of them.
The story begins on a TV game show entitled "Peeping Toms" which is extreme enough to be a Monty Python spoof (if done 'honestly,') and yet, in current reality TV terms also a practical, workable concept (if set up as 'reality' TV is set up, that is, to have a loose preliminary script, to use professional hopefuls, to cover all legal aspects beforehand, etc.)
An apparently blind girl goes into the men's locker and starts undressing: Will the surprise contestant be revealed to be chivalrous or a cad?
As it turns out, the guy is not only chivalrous, but proves to be a nice guy overall, he gets a free dinner, she gets a cutlery prize set. They go out to dinner together, she stabs him with the cutlery set and kills him.
The murder is witnessed by a neighbor, the police are called in, the crime scene is cleared (by a creepy looking, stalker, ex-husband) by the time the police get there and the rest of the movie is devoted to proving that the murder was committed and that Danielle and her ex-husband are a danger to society.
This is a pretty straightforward story. The devil is in the details and in the weird turns we take on our way to the conclusion.
For example, the guy is Black. The dinner he wins is in The African Room, a themed restaurant prominently featuring a stuffed gorilla, and then we get a story about Danielle (a French Canadian with a thick, distracting Quebecois accent) being alone on top of the Empire State Building. Is this all some kind of outlandish King Kong reference? For what possible purpose and to what end?
The murder witness has some credibility problems. She is a reporter whose had some clashes with police authorities and when recognized is not immediately believed, something which is exacerbated when no evidence of her statements is found (or is inadvertently destroyed.)
She, with help of a private investigator then look into the crime, with following the couple and her ending up eventually at an experimental mental institution, where she is recognized by the killers. She is drugged and hypnotized and the details of the plot revealed to her while under that state, so that what we see is a confusing, hallucinatory phantasmagoria where she is cast in the part of the sister.
Danielle is a disturbed, (recently) separated Siamese twin who suffers from dating 'problems': All her dates remind her of her doctor ex-husband, who was responsible for the death of her sister Dominique during the separation surgery. Danielle partly shares the guilt as she understandably, desired intimacy and privacy and expressed that something needed to be done.
By the end of the sequence Danielle's psychosis is triggered again (with the reporter present, still drugged,) and she attacks and kills the doctor.
The reporter, under the effects of post-hypnotic suggestion denies that there ever was a murder.
...In the meantime, the investigator still trails the murdered body of that initial guy.
The overall effect ends up being playful and somewhat humorous, and it seems that De Palma treats it all with a light touch.
The critical references to voyeurism do not feel jaded and mean spirited as they do by the time we get to Dressed To Kill.
The psychological aspects remain of primary importance despite the implication that biological duality or multiplicity have anything whatsoever to do with the story.
The ultimate point of the film remains confusing though, ending closer in territory to something straightforward like Jekyll/Hyde than its visuals would imply with all its imagery of fetuses (during the credits,) Siamese twins (during the TV documentary,) triplets, etc. (during the hallucination sequence.)
Visually and stylistically, however, all that twin imagery is extremely enriching to the film watching experience (as The Manster, The Dark Half& Army of Darkness have also proven.)
It's a trip, for sure.
Check it out.