  We study cyclic sieving phenomena (CSP) on combinatorial objects from an
abstract point of view by considering a rational polyhedral cone determined by
the linear equations that define such phenomena. Each lattice point in the cone
corresponds to a non-negative integer matrix which jointly records the
statistic and cyclic order distribution associated with the set of objects
realizing the CSP. In particular we consider a universal subcone onto which
every CSP matrix linearly projects such that the projection realizes a CSP with
the same cyclic orbit structure, but via a universal statistic that has even
distribution on the orbits.
  Reiner et.al. showed that every cyclic action give rise to a unique
polynomial (mod $q^n-1$) complementing the action to a CSP. We give a necessary
and sufficient criterion for the converse to hold. This characterization allows
one to determine if a combinatorial set with a statistic give rise (in
principle) to a CSP without having a combinatorial realization of the cyclic
action. We apply the criterion to conjecture a new CSP involving stretched
Schur polynomials and prove our conjecture for certain rectangular tableaux.
Finally we study some geometric properties of the CSP cone. We explicitly
determine its half-space description and in the prime order case we determine
its extreme rays.
