A separation algorithm is a procedure for generating cutting planes. Up to now, only a few polynomial-time separation algorithms were known for the Boolean quadric and cut polytopes. These polytopes arise in connection with zero–one quadratic programming and the max-cut problem, respectively. We present a new algorithm, which separates over a class of valid inequalities that includes all odd bicycle wheel inequalities and (2p+1,2)(2p+1,2)-circulant inequalities. It exploits, in a non-trivial way, three known results in the literature: one on the separation of View the MathML source{0,12}-cuts, one on the symmetries of the polytopes in question, and one on an affine mapping between the polytopes.