The subgroup structure suggests another way to compose an arbitrary isometry:
: Pick a fixed point, and a mirror through it.
# If the isometry is odd, use the mirror; otherwise do not.
# If necessary, rotate around the fixed point.
# If necessary, translate.
This works because translations are a normal subgroup of the full group of isometries, with quotient the orthogonal group; and rotations about a fixed point are a normal subgroup of the orthogonal group, with quotient a single reflection.