text: Composition of isometries mixes kinds in assorted ways. We can think of the identity as either two mirrors or none; either way, it has no effect in composition. And two reflections give either a translation or a rotation, or the identity (which is both, in a trivial way). Reflection composed with either of these could cancel down to a single reflection; otherwise it gives the only available three-mirror isometry, a glide reflection. A pair of translations always reduces to a single translation; so the challenging cases involve rotations. We know a rotation composed with either a rotation or a translation must produce an even isometry. Composition with translation produces another rotation (by the same amount, with shifted fixed point), but composition with rotation can yield either translation or rotation. It is often said that composition of two rotations produces a rotation, and Euler proved a theorem to that effect in 3D; however, this is only true for rotations sharing a fixed point.