  We consider the problem of extending an acyclic binary relation that is
invariant under a given family of transformations into an invariant preference.
We show that when a family of transformations is commutative, every acyclic
invariant binary relation extends. We find that, in general, the set of
extensions agree on the ranking of many pairs that (i) are unranked by the
original relation, and (ii) cannot be ranked by invariance or transitivity
considerations alone. We interpret these additional implications as the
out-of-sample predictions generated by invariance, and study their structure.
