Abstract. 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.