  For two elements $v$ and $w$ of the symmetric group $\mathfrak{S}_n$ with
$v\leq w$ in Bruhat order, the Bruhat interval polytope $Q_{v,w}$ is the convex
hull of the points $(z(1),\ldots,z(n))\in \mathbb{R}^n$ with $v\leq z\leq w$.
It is known that the Bruhat interval polytope $Q_{v,w}$ is the moment map image
of the Richardson variety $X^{v^{-1}}_{w^{-1}}$. We say that $Q_{v,w}$ is
\emph{toric} if the corresponding Richardson variety $X_{w^{-1}}^{v^{-1}}$ is a
toric variety. We show that when $Q_{v,w}$ is toric, its combinatorial type is
determined by the poset structure of the Bruhat interval $[v,w]$ while this is
not true unless $Q_{v,w}$ is toric. We are concerned with the problem of when
$Q_{v,w}$ is (combinatorially equivalent to) a cube because $Q_{v,w}$ is a cube
if and only if $X_{w^{-1}}^{v^{-1}}$ is a smooth toric variety. We show that a
Bruhat interval polytope $Q_{v,w}$ is a cube if and only if $Q_{v,w}$ is toric
and the Bruhat interval $[v,w]$ is a Boolean algebra. We also give several
sufficient conditions on $v$ and $w$ for $Q_{v,w}$ to be a cube.
