We use the invariant expansion of an effective Hamiltonian to investigate exciton-spin dephasing and relaxation in a simple two-band bulk semiconductor. When compared to the situation at the Γ point, the point group symmetry of a crystal can be broken at finite wave vectors. As a consequence, depending on the propagation direction, the exchange interaction between electrons and holes in conjunction with the center of mass motion of the excitons can mix different states. This leads to spin beating of excitons. Here we consider exchange terms which are linear, quadratic, and cubic in the wave vector Q and their effect for different directions of propagation. Since dipole active excitons have either transverse or longitudinal character, their total angular momentum (pseudospin) has to be defined with respect to the direction of propagation of the quasiparticles. Therefore, scattering processes, in which the direction of Q is changed, result in a variation of the quantization direction and lead to spin relaxation. In low symmetry directions, both processes contribute to the time evolution of the spin state.