In recent years, the interplay between symmetry and rigidity has received considerable attention. For example, it is well-known that the rigidity matrix for a finite bar-joint framework with an abelian symmetry group admits a block-diagonalisation over the irreducible representations of the group. Moreover, the diagonal blocks can be described explicitly by associated orbit matrices. It is also known that the rigidity matrix for a periodic bar-joint framework gives rise to a Hilbert space operator which is unitarily equivalent to a multiplication operator with a matrix-valued symbol function. This symbol function determines the RUM spectrum and the phenomenon of rigid unit modes (RUMs). RUM theory for periodic frameworks and the aforementioned decomposition theory for finite symmetric frameworks can be viewed as two sides of the same coin. The first aim of this talk is to formalise this viewpoint using techniques from Fourier analysis. The second aim is to extend the theory so that it may be applied in new contexts. This is joint work with Eleftherios Kastis and John McCarthy.