  We prove a variant of the well-known result that intertwiners for the
bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication
operators on $L^2(T)$. This enables us to unify and extend fundamental aspects
of rigidity theory for bar-joint frameworks with an abelian symmetry group. In
particular, we formulate the symbol function for a wide class of frameworks and
show how to construct generalised rigid unit modes in a variety of new
contexts.
