An affine map between two polyhedral complexes which is bijective on the underlying sets need not induce an isomorphism of the complexes preserving the induced cell-complex structure. The key observation in this paper is that this is nevertheless true under an additional assumption: If an affine bijection \(f: T_1\to T_2\) between two polyhedral complexes \(T_1, T_2\), both of which consist of a union of faces of two convex polyhedra \(P_1\), \(P_2\), extends to an affine map from \(P_1\) to \(P_2\) then \(f\) necessarily respects the cell-complex structure of \(T_1\) and \(T_2\) induced by \(P_1\) and \(P_2\), respectively. This result depends on certain properties of extremal points and extremal subsets of polyhedra. There are applications in the sciences, including biology and molecular evolution.