I define the equivalence relation as HJRW defines them (that is twin vertices may be adjacent).  In this way, each equivalence class is either a clique or a stable set (this is what you mention in the comments too).  Thus, the number of such equivalence classes gives an upper bound on the [cochromatic number](http://en.wikipedia.org/wiki/Cocoloring) $z(G)$ of $G$.  The cochromatic number of $G$ is the minimum number of colours needed to colour $V(G)$ such that each colour class induces a clique or stable set.