Inverse Optimal Transport (IOT) studies the problem of inferring the underlying cost that gives rise to an observation on coupling two probability measures. Couplings appear as the outcome of matching sets (e.g. dating) and moving distributions (e.g. transportation). Compared to Optimal transport (OT), the mathematical theory of IOT is undeveloped. We formalize and systematically analyze the properties of IOT using tools from the study of entropy-regularized OT. Theoretical contributions include characterization of the manifold of cross-ratio equivalent costs, the implications of model priors, and derivation of an MCMC sampler. Empirical contributions include visualizations of cross-ratio equivalent effect on basic examples, simulations validating theoretical results and experiments on real world data.