Suppose that you've got a logistic regression with multiple nominal outcomes that cannot be ordered in a theoretically meaningful way. Assume further, however, that the proportional odds assumption nonetheless holds true for these categories for a given order. In this scenario, would it be possible to fit an ordered logistic regression model?
What I'm really asking is if there's any reason, apart from the proportional odds assumption, for why a ordered logit needs ordered categories? Is it simply a mathematical assumption or is there something else to it that I'm presently missing?