Anybody know the name of this film?

There is a film I want, but I can’t remember the title. It is French made, late ‘50s or early ‘60s, black and white. The film is an adaptation of a controversial book published in France in the 1920s or ‘30s. The author of the book and director of the film are quite famous. The story is about a brother and sister living in Paris during the time the book was written, who find themselves orphaned and the older sister takes care of her younger brother. But early on, they start an emotion-based game, a kind of competition that neither will lose at any cost. They are necessarily close, but this game becomes more serious as time goes on. Identity exchanges occur, incest, etc. Eventually, as the boy gets older, another girl enters their lives and the relationship with the siblings come to a head. It ends with one of them getting sick and bed-ridden, and I believe there is a mercy killing and a suicide and the other girl commits suicide or something. It’s supposed to be a cultural classic, but I’ve never seen it, only stills from it. II figure you’d know the name of the book which is the same as the film.