  The framework of spherical transforms and P\'olya ensembles is of utility in
deriving structured analytic results for sums and products of random matrices
in a unified way. In the present work, we will carry over this framework to
study products of unitary matrices. Those are not distributed via the Haar
measure, but still are drawn from distributions where the eigenvalue and
eigenvector statistics factorise. They include the circular Jacobi ensemble,
known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz
determinants, as well as the heat kernel for Brownian motion on the unitary
group. We define cyclic P\'olya frequency functions and show their relation to
the cyclic P\'olya ensembles, give a uniqueness statement for the corresponding
weights, and derive the determinantal point processes of the eigenvalue
statistics at fixed matrix dimension. An outline is given of problems one may
encounter when investigating the local spectral statistics.
