  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.
