The Other, and probably because a) I saw it when I was very little back in the 70's (it was released in 1972, I believe, directed by Robert Mulligan of To Kill a Mockingbird fame) and b) because I have a twin brother.
The move stayed with me for a very, very long time. So much so that, in the mid 90's when I was in college, someone asked the professor pretty much the same question posed here and he replied he couldn't remember the movie's title, but described the events and I knew it was the same movie (I, myself, had forgotten its title.)
Less than a year later, as I perused Leonard Maltin's Movie Guide - EVERY page - looking for movies to watch, I came across the title and knew immediately this was the one. I was fortunate to find it a year or two later in a movie rental place and then it was subsequently released on DVD a few years later.
I don't want to mention anything about the plot, but if you got to amazon.com and read reviews, you'll see the movie left quite a few people disturbed. It's "quiet" horror, and if I'm not mistaken, was rated PG at the time of its release though it may be PG-13 now. Great acting and period piece as one might expect from Robert Mulligan, but man, the story is just creepy - based on the Thomas Tryon novel.