Parity-odd polyspectra of spatial point processes

We study how handedness enters the factorial cumulants and polyspectra of stationary, properly isotropic spatial point processes. Rotational symmetry forces parity-odd cumulants to vanish at every order up to the spatial dimension. At the next order, the odd part of the factorial moment is necessarily connected. We show that this threshold is sharp by constructing a relabelling-invariant measure of simplex handedness and using it to define a chiral Poisson-cluster process. The resulting factorial polyspectrum has a nonzero parity-odd component, which changes sign under reflection and cancels when the two mirror laws are mixed equally.