An effective model of the hexagonal (NiAs-structure) manganese telluride valence band in the vicinity of the A point of the Brillouin zone is derived. It is shown that whereas for the usual antiferromagnetic order (magnetic moments on the basal plane) band splitting at A is small, their out-of-plane rotation enhances the splitting dramatically (to about 0.5 eV). We propose extensions of recent experiments [Phys. Rev. Mater. 6, 014404 (2022)] where such inversion of magnetocrystalline anisotropy has been observed in Li-doped MnTe to confirm this unusual sensitivity of a semiconductor band structure to magnetic order.