There are some neat patterns I've noticed.
If you take a tetrahedron with it's dual as itself, you can insert triangular faces of the dual at each vertex of the original tetrahedron and rotate the original faces so as to get a new polyhedron. In fact, you get an octahedron!
If you repeat this idea again, you can take the dual of the octahedron, a cube, and place squares at the vertices of the octahedron and rotate the triangular faces of the octahedron to get a cuboctahedron.
The icosidodecahedron gives a striking example of what I'm talking about by combining the mutual duals the icosahedron and the dodecahedron.
Question: Does this process have a name? I'll admit I don't have this process precisely defined. Is there some form of this process that always works for polyhedra with non-regular duals, for instance, some of the Archimedean solids? For instance, I'd love to learn the name of some sort of polyhedron that combines the dual of a rhombicuboctahedron with itself.
The operation you're describing is called rectification, and can be understood as truncating "all the way". It's defined for all polytopes with respect to their dual.
If you continue the process beyond this point, you will effectively see a reversal of truncation process towards the dual polytope. This is called bitruncation, or birectification if you truncate "all the way" twice.