Physical phenomena other than gravity are customarily assumed to be described by Lorentz-covariant theories, and the validity of the Lorentz transformation has been empirically verified to very high accuracy. But if all nongravitational phenomena really are Lorentz covariant, it would challenge physical consistency for gravity not to be Lorentz covariant as well. Here we work out the Lorentz-covariant dynamics of test bodies that interact with, respectively, any electromagnetic four-vector potential and any gravitational symmetric-second-rank-tensor potential. A subsidiary symmetry spontaneously emerges in each case: gauge invariance in electromagnetism, and general coordinate covariance in gravitation. We then work out field equations for the electromagnetic and gravitational potentials which incorporate their respective subsidiary symmetries; such field equations unavoidably have an infinite number of candidate solutions for their potentials, but candidate potentials which aren’t Lorentz covariant are of course excluded, as are candidate potentials which violate causality or introduce singularities.