  In this paper we examine robust clustering behaviour with multiple nontrivial
clusters for identically and globally coupled phase oscillators. These systems
are such that the dynamics is completely determined by the number of
oscillators N and a single scalar function $g(\varphi)$ (the coupling
function). Previous work has shown that (a) any clustering can stably appear
via choice of a suitable coupling function and (b) open sets of coupling
functions can generate heteroclinic network attractors between cluster states
of saddle type, though there seem to be no examples where saddles with more
than two nontrivial clusters are involved. In this work we clarify the
relationship between the coupling function and the dynamics. We focus on cases
where the clusters are inequivalent in the sense of not being related by a
temporal symmetry, and demonstrate that there are coupling functions that give
robust heteroclinic networks between periodic states involving three or more
nontrivial clusters. We consider an example for N=6 oscillators where the
clustering is into three inequivalent clusters. We also discuss some aspects of
the bifurcation structure for periodic multi-cluster states and show that the
transverse stability of inequivalent clusters can, to a large extent, be varied
independently of the tangential stability.
