We consider a hybrid structure where a material with Rashba-like spin-orbit coupling is proximity coupled to a conventional superconductor. We find that the superconducting critical temperature T c can be tuned by rotating the vector n characterizing the axis of broken inversion symmetry. This is explained by a leakage of s-wave singlet Cooper pairs out of the superconducting region, and by conversion of s-wave singlets into other types of correlations, among these s-wave odd-frequency pairs robust to impurity scattering. These results demonstrate a conceptually different way of tuning T c compared to the previously studied variation of T c in magnetic hybrids.