The context you are citing seems to be using the phrase "reverses containment" in a problematic way.
The normal meaning of "$f$ reverses containment" is this: if $x\subseteq y$, then $f(x)\supseteq y$". Really we are just talking about a function reversing a partial order, of which containment is an example. So the first problem is that I do not see any sensible concept of "partial order" for the faces of a polygon.
Apparently, then, this interpretation is out the window.
Another candidate is that it rather refers to the placement of the polyhedron after applying the duality transformation. The idea is that if you let $f$ associate each face of a cube with its midpoint, you wind up with a set of points forming the vertices of a octahedron inside the cube. Applying an analogous mapping to the octahedron, you get a smaller cube inside the octahedron, not the same cube you started with. But we could identify these polytopes with each other.
Still, this interpretation is problematic with the rest of the context, because it does not furnish a map between $S(C)$ and $S(O)$.
Finally, the other likely candidate is the "containment" of the central point of a face inside the face itself. A better word would be "membership" rather than containment. This is also reasonable, but it still suffers the conflict with the context that it is not a map from $S(C)\to S(O)$. Still, this interpretation is what I believe is the normal concept of the duality of these two polyhedra (and of duality between other pairs.) As far as I can see, the author just misexpressed the description of $f$.
If I were attempting to reconstruct a plausible version of the exposition you are describing, I would guess it would be this:
Let $C$ be a cube and $O$ be an octahedron, and $V(P)$ be the set of vertices of polytope $P$ and $F(P)$ be the set of faces of the polytope $P$.
Then $C$ and $O$ are dual in the sense that there is a special bijection $f:F(C)\to V(O)$, mapping each face to its center with the property that $f(F_1)\in F_1$ for any face $F_1\in F(C)$, and there is, likewise, a bijection $g:F(O)\to V(C)$ with analogous properties.
Alternatively, if the "polytope containment" interpretation seems more likely, you could say that $f$ is a function from the set of cubes in $\mathbb R^3$ to the set of octahedrons in $\mathbb R^3$, where $f$ sends a cube to the octahedron it contains.