We consider the problem of testing the rationalizability of choice data by a preference satisfying an arbitrary collection of <em>invariance</em> axioms.  Examples of such axioms include quasilinearity, homotheticity, independence-type axioms for mixture spaces, constant risk and ambiguity aversion axioms, stationarity, separability, and many others. We provide necessary and sufficient conditions for invariant rationalizability via a novel approach which relies on tools from the theoretical computer science literature on automated theorem proving.   We also establish a generalization of the Dushnik-Miller theorem, which we use to give a complete description of the counterfactual predictions generated by the data under any such collection of axioms.