Given a poset $S$, one can form a new poset $I(S)$ whose elements are intervals in $S$ (i.e. either $\emptyset$ or $[a,b]$ for some $a\leq b\in S$) with ordering by (set) inclusion. If $S$ is ranked, then $I(S)$ will also be ranked (by $r([a,b])=r(b)-r(a)$).
If $S$ is the face lattice of a $d$-dimensional polytope $P$, is there a canonical way to construct a $d+1$-dimensional polytope $I(P)$ with face lattice $I(S)$? Is there a name for this construction?
1) The 2-faces will always be quadrilaterals.
2) The underlying cellular complex is not the barycentric subdivision, whose faces are the chains of $S$, not the intervals.
3) If you apply the construction to a simplex, you should get a cube (of one higher dimension).
4) Of course the best construction should preserve symmetries and intertwine the inclusion of a face $F$ into $P$ with that of $I(F)$ into $I(P)$.
5) The only polytope I actually need an answer for right now is the regular 3-dimensional cube. If this construction only works for, say, simple polytopes, I'm fine with that.