Generalizations
In higher dimensions, any rigid motion that preserve a point in dimension 2n or 2n+1 is a composition of at most n rotations in orthogonal planes of rotation, though these planes need not be uniquely determined, and a rigid motion may fix multiple axes.
A rigid motion in 3 dimensions that does not necessarily fix a point is a "screw motion". This is because a composition of a rotation with a translation perpendicular to the axis is a rotation about a parallel axis, while composition with a translation parallel to the axis yields a screw motion; see screw axis. This gives rise to screw theory.
Read more about this topic: Euler's Rotation Theorem