  The Watanabe-Strogatz and Ott-Antonsen theories provided a seminal framework
for rigorous and comprehensive studies of collective phenomena in a broad class
of paradigmatic models for ensembles of coupled oscillators. Recently, a
"circular cumulant" approach was suggested for constructing the perturbation
theory for the Ott-Antonsen approach. In this paper, we derive the relations
between the distribution of Watanabe-Strogatz phases and the circular cumulants
of the original phases. These relations are important for the interpretation of
the circular cumulant approach in the context of the Watanabe-Strogatz and
Ott-Antonsen theories. Special attention is paid to the case of hierarchy of
circular cumulants, which is generally relevant for constructing perturbation
theories for the Watanabe-Strogatz and Ott-Antonsen approaches.
