Minkowskian spacetime crystals were recently introduced as time-variant systems described by constitutive relations that are observer independent. Here, we prove that the most general class of materials, invariant under a Lorentz boost along a fixed spatial direction is composed of biaxial crystals, where one of the optical axes of the permittivity and permeability tensor aligns parallel to the direction of the modulation velocity. We elucidate how to take advantage of the unique properties of this class of time-variant systems, to characterize in an expedite manner the wave propagation in such systems.