After reviewing the foundations of special relativity and the room left for rival theories, a set of nonrelativistic para-Lorentzian transformations is derived uniquely, based on (a) a weaker first principle, (b) the requirement that the transformations sought do not give rise to the clock “paradox” (in a refined version), and (c) the compliance of the transformations with the classical experiments of Michelson-Morley, Kennedy-Thorndike, and Ives-Stilwell. The corresponding dynamics is developed. Most of the experimental support of special relativity is reconsidered in (...) the light of the new theory. It is concluded that the relativity of simultaneity has so far not been tested. (shrink)