The predictions of local hidden variable models for projective measurements on qubits correspond to binary colourings of the Bloch sphere with antipodal points oppositely coloured. We consider here the predictions of quantum theory and local hidden variables for the correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by a given angle \theta. We motivate and explore the Hemispherical Colouring Maximality Hypothesis (HCMH): that, for a continuous range of \theta > 0, the maximum anti-correlation is obtained by assigning to one qubit the colouring with one hemisphere black and the other white, and assigning the reverse colouring to the other. If provable, this would produce an infinite set of Bell inequalities separating quantum and classical correlations. We describe numerical tests that are consistent with the HCMH and bound the range of \theta. We also note proofs of related results for binary colourings of R^n.