Most school mathematics courses nowadays include a study of symmetry in the euclidean sense—isometries in 2- or 3-dimensional euclidean space which map a figure to itself. This article explores the theme of symmetry in a more general, topological sense. It could form the basis of a short self-contained unit in topology with a worthwhile end-point, which teachers might care to try out with students at sixth form level or perhaps in a college of education course.