A reinterpretation is the new reading, the new showing, or the new performance itself — the result of thinking about something, showing it, or playing it in a new way. You can count it: her reinterpretation of the song, a bold reinterpretation of the play. You can also use it as a general name for the work of seeing an old thing freshly: the film is a reinterpretation of a folk tale. Art reviews, history books, and law talk use it a lot, because they need a name for the new version, not only the verb. Though actually, it still points back to the old work. A sequel is a new story that follows on; a reinterpretation is the same story (or the same facts, or the same song) understood or presented differently.