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We classify all quotients $W/W_J$ up to isomorphism in Bruhat order, with$(W,S)$ a Coxeter system and $W_J$ a parabolic subgroup of $W$. In particular,the non-trivial isomorphisms fall into a small number of cases which are highlyrestricted; all have $W$ finite and $W_J$ a maximal parabolic. This has theimmediate application of classifying dominant and antidominant blocks ofcategory $\mathcal O$ for Kac-Moody algebras up to equivalence.

Classifying quotients on Coxeter groups by isomorphism in Bruhat order