  Equivalence relations on the edge set of a graph $G$ that satisfy restrictive
conditions on chordless squares play a crucial role in the theory of Cartesian
graph products and graph bundles. We show here that such relations in a natural
way induce equitable partitions on the vertex set of $G$, which in turn give
rise to quotient graphs that can have a rich product structure even if $G$
itself is prime.
