The authors investigate the effect of the microscopic potential on the conduction band wavefunctions in III-V superlattices under hydrostatic pressure. In particular, they show that the pressure dependence of the cross interface luminescence in GaAs-AlAs structures with perfect and imperfect interfaces exhibits fine details of the valley mixing, which changes the strength of the luminescence signal over many orders of magnitude. They predict selection rules which can be used to turn pressure-induced cross interface luminescence into a sensitive tool for the characterization of the microscopic signature of the interface quality.