Swinton added, “A psychoanalyst, or a psychiatrist with a sense of the unconscious, is someone who knows that in every delusion is an attempt to tell a truth.” Noting Klemperer’s preoccupation with a spouse he had lost in the war, she said, “Klemperer is inhabited by the phantasm of his lost wife: He is, in this crucial respect, ‘played’ by a woman. She dictates the rhythm of his life in the everyday texture of his bereaved loneliness.”

And while to say much more would be a spoiler, there is also a third, more monstrous character that Swinton plays in the final act, and Guadagnino intentionally conceived these three figures for her. “This is a movie that is very connected to psychoanalysis,” he said, “and I like to think that only Tilda could play ego, superego and id.”

To aid Swinton in her transformation into Klemperer, Guadagnino hired the Oscar-winning makeup artist Mark Coulier. Though Coulier had previously put Swinton in old-age prosthetics for “The Grand Budapest Hotel,” and Swinton has played male characters before on film (most notably, in the 1992 “Orlando”) to do both at the same time would be a new challenge. In fact, the striking features that have served Swinton well in other transformations were of no use as Klemperer.

“Although she has a slightly androgynous look from sort of a fashion-model point of view, Tilda’s got a very feminine bone structure,” said Coulier, who thickened Swinton’s neck with prosthetics and built her jaw out to look heavier and more masculine.

The makeup process took up to four hours each day, and Swinton spent more days in heavy makeup for “Suspiria” than she had for any other project. Still, the actress kept her spirits up by requesting that Coulier fashion her a different set of prosthetics.

“She did have us make a penis and balls,” Coulier said. “She had this nice, weighty set of genitalia so that she could feel it dangling between her legs, and she managed to get it out on set on a couple of occasions.” And where is Swinton’s superfluous genitalia now? “Probably in a box somewhere!” Coulier said brightly. “I should try and find it, and put it on a plaque on the wall of my workshop.”