  We study the phenomenon of cluster synchrony that occurs in ensembles of
coupled phase oscillators when higher-order modes dominate the coupling between
oscillators. For the first time, we develop a complete analytic description of
the dynamics in the limit of a large number of oscillators and use it to
quantify the degree of cluster synchrony, cluster asymmetry, and switching. We
use a variation of the recent dimensionality-reduction technique of Ott and
Antonsen [Chaos {\bf 18}, 037113 (2008)] and find an analytic description of
the degree of cluster synchrony valid on a globally attracting manifold. Shaped
by this manifold, there is an infinite family of steady-state distributions of
oscillators, resulting in a high degree of multi-stability in the cluster
asymmetry. We also show how through external forcing the degree of asymmetry
can be controlled, and suggest that systems displaying cluster synchrony can be
used to encode and store data.
