Domain adaptation algorithms and theory have relied upon an assumption that the observed data uniquely specify the correct correspondence between the domains. Unfortunately, it is unclear under what conditions this identifiability assumption holds, even when restricting ourselves to the case where a correct bijective map between domains exists. We study this bijective domain mapping problem and provide several new sufficient conditions for the identifiability of linear domain maps. As a consequence of our analysis, we show that weak constraints on the third moment tensor suffice for identifiability, prove identifiability for common latent variable models such as topic models, and give a computationally tractable method for generating certificates for the identifiability of linear maps. Inspired by our certification method, we derive a new objective function for domain mapping that explicitly accounts for uncertainty over maps arising from unidentifiability. We demonstrate that our objective leads to improvements in uncertainty quantification and model performance estimation.