It is well known that (i) the flexibility and rigidity of proteins are central to their function, (ii) a number of oligomers with several copies of individual protein chains assemble with symmetry in the native state and (iii) added symmetry sometimes leads to added flexibility in structures. We observe that the most common symmetry classes of protein oligomers are also the symmetry classes that lead to increased flexibility in certain three-dimensional structures—and investigate the possible significance of this coincidence. This builds on the well-developed theory of generic rigidity of body–bar frameworks, which permits an analysis of the rigidity and flexibility of molecular structures such as proteins via fast combinatorial algorithms. In particular, we outline some very simple counting rules and possible algorithmic extensions that allow us to predict continuous symmetry-preserving motions in body–bar frameworks that possess non-trivial point-group symmetry. For simplicity, we focus on dimers, which typically assemble with twofold rotational axes, and often have allosteric function that requires motions to link distant sites on the two protein chains.