'You can't come down here! No, sorry, you're going to have to go round by the swivelling staircase, someone's let off Garrotting Gas just along here -'
--Ginny Weasley diverting students from the corridor where Harry was about to break into Umbridge's office
The swivelling staircase is a magically enchanted staircase in Hogwarts. It is located in the vicinity of the Defense Against the Dark Arts teacher’s office on the second floor (OP32).
This is the only mention of a swivelling or rotating staircase in canon. The film shows a vast, cavernous hall with numerous rotating staircases, but these don't exist in the books.