Unsupervised domain translation (UDT) aims to find functions that convert samples from one domain (e.g., sketches) to another domain (e.g., photos) without changing the high-level semantic meaning (also referred to as "content"). The translation functions are often sought by probability distribution matching of the transformed source domain and target domain. CycleGAN stands as arguably the most representative approach among this line of work. However, it was noticed in the literature that CycleGAN and variants could fail to identify the desired translation functions and produce content-misaligned translations.This limitation arises due to the presence of multiple translation functions---referred to as ``measure-preserving automorphism" (MPA)---in the solution space of the learning criteria. Despite awareness of such identifiability issues, solutions have remained elusive. This study delves into the core identifiability inquiry and introduces an MPA elimination theory. Our analysis shows that MPA is unlikely to exist, if multiple pairs of diverse cross-domain conditional distributions are matched by the learning function.Our theory leads to a UDT learner using distribution matching over auxiliary variable-induced subsets of the domains---other than over the entire data domains as in the classical approaches. The proposed framework is the first to rigorously establish translation identifiability under reasonable UDT settings, to our best knowledge.Experiments corroborate with our theoretical claims.