Physical phenomena, sometimes with the exception of gravity, are usually assumed to be described by Lorentz transformation covariant theories, and the validity of the Lorentz transformation has been empirically verified to very high accuracy. The Einstein equation of gravity theory, however, has an infinite set of metric solutions, an infinite subset of which aren't Lorentz covariant, and one of the latter might be taken as valid, e.g., the Robertson-Walker metric for cosmology. But if all of nongravitational physics is in fact Lorentz covariant, it would almost certainly be physically inconsistent for gravity theory not to be Lorentz covariant as well. The solution ambiguity of the Einstein equation is a consequence of its important symmetry of general coordinate transformation covariance. However the four-vector potential form of electromagnetic theory has an analogous solution ambiguity as a consequence of its important symmetry of gauge transformation invariance, but in that case it is standard practice to break this symmetry by imposing the retarded Lorentz gauge condition, the simplest gauge condition which is Lorentz covariant and causal. Here we show that both gauge transformation invariance in electromagnetic theory and general coordinate transformation covariance in gravity theory arise spontaneously from fully Lorentz covariant initial assumptions. These subsidiary dynamic symmetries crucially affect the structure of the equations of their respective theories, but any solutions they happen to admit which aren't fully Lorentz covariant are ipso facto excluded by the fully Lorentz covariant initial assumptions.