InₓGa₁₋ₓN structures epitaxially grown on a-plane or m-plane GaN exhibit in-plane optical polarization. Linear elasticity theory treats the two planes equivalently and is hence unable to explain the experimentally observed higher degree of linear polarization for m-plane than a-plane InₓGa₁₋ₓN. Using density functional theory, we study the response of InₓGa₁₋ₓN random alloys to finite biaxial strains on both non-polar planes. The calculated m-plane InₓGa₁₋ₓN valence band splitting is larger than that of a-plane, due to a greater degree of structural relaxation in a-plane InₓGa₁₋ₓN. We provide a parameterization of the valence band splitting of InₓGa₁₋ₓN strained to a-plane and m-plane GaN for In compositions between 0 and 0.5, which agrees with experimental measurements and qualitatively explains the experimentally observed difference between a-plane and m-plane polarization.