Self-dual intervals in the Bruhat order

Bj\"orner-Ekedahl prove that general intervals $[e,w]$ in Bruhat order are "top-heavy", with at least as many elements in the $i$-th corank as the $i$-th rank. Well-known results of Carrell and of Lakshmibai-Sandhya give the equality case: $[e,w]$ is rank-symmetric if and only if the permutation $w$ avoids the patterns $3412$ and $4231$ and these are exactly those $w$ such that the Schubert variety $X_w$ is smooth. In this paper we study the finer structure of rank-symmetric intervals $[e,w]$, beyond their rank functions. In particular, we show that these intervals are still "top-heavy" if one counts cover relations between different ranks. The equality case in this setting occurs when $[e,w]$ is self-dual as a poset; we characterize these $w$ by pattern avoidance and in several other ways.