We describe a new Zome-like system that exhibits octahedral rather than icosahedral symmetry, and illustrate its application to 3-dimensional projections of 4-dimensional regular polychora. Furthermore, we explain the existence of that system, as well as an infinite family of related systems, in terms of Hamilton’s quaternions and the binary icosahedral group. Finally, we describe a remarkably tenacious aspect of H4 symmetry that “survives” projection down to three dimensions, reappearing only in 2-dimensional projections.