Using anisotropic elasticity theory, we analyze the relative thermodynamic stabilities of strained graphitic (hexagonal) BN and cubic BN (cBN) single-crystal structures for all orientations ofbiaxial stress and strain fields relative to the crystallographic directions. In hBN, the most thermodynamically stable orientation has the graphitic basal planes oriented roughly 45 degrees relative to either the plane of stress or strain. For cBN, the lowest-energy configuration differs for the constant stress or constant strain assumptions. Importantly, these most-stable orientations of hBN and cBN differ from those found experimentally for graphitic BN and cBN in polycrystalline BN films produced by energetic deposition processes. Therefore, the observed textures are not those that minimize elastic strain energy. We discuss possible origins other than elastic strain-energy effects for the observed textures.