Classifying quotients on Coxeter groups by isomorphism in Bruhat order

We classify all quotients $W\mathrm{/}{W}_J$ 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 highly restricted; all have $W$ finite and $W_J$ a maximal parabolic. This has the immediate application of classifying dominant and antidominant blocks of category $\mathcal{O}$ for Kac-Moody algebras up to equivalence.