If the facets of a polytope $A$ are in one-to-one correspondence with the members of a finite set $X$ and the facets of a polytope $B$ are also in one-to-one correspondence with the members of $X$ (ie. both polytopes have the same number of facets), under what circumstances are we able to state that the two polytopes are isomorphic?
Is it sufficient if $A$ and $B$ both have the same number of vertices as well as number of facets, as in this case there will be a mapping sending each vertex of $A$ to a vertex of $B$.