  Actions of $\mathbb{Z}^d$ by automorphisms of compact zero-dimensional groups
exhibit a range of mixing behaviour. Schmidt introduced the notion of mixing
shapes for these systems, and proved that non-mixing shapes can only arise
non-trivially for actions on zero-dimensional groups. Masser has shown that the
failure of higher-order mixing is always witnessed by non-mixing shapes. Here
we show how valuations can be used to understand the (non-)mixing behaviour of
a certain family of examples. The sharpest information arises for systems
corresponding to tight polyhedra.
