Using health as an example, we consider comparing two latent distributions when only ordinal data are available. Distinct from the literature, we assume a continuous latent distribution but not a parametric model. Primarily, we contribute (partial) identification results: given two known ordinal distributions, what can be learned about the relationship between the two corresponding latent distributions? Secondarily, we discuss Bayesian and frequentist inference on the relevant ordinal relationships, which are combinations of moment inequalities. Simulations and empirical examples illustrate our contributions.