  We give a thorough analysis of those subgroups of SO(3) generated by
rotations about perpendicular axes by 2\pi/p and 2\pi/q. A corollary is that
such a group is the free product of the cyclic groups of rotations about the
separate axes if and only if p,q\ge 3 and are both odd. These groups are
naturally associated with a family of hierarchical tilings of Euclidean
3-space.
