I had done something similar a few years ago. It is available in mandelbulber as "generalized box fold".

http://www.fractalforums.com/amazing-box-amazing-surf-and-variations/generalized-box-fold/

I think the two approaches have similarities, but they modify different parts of the Mandelbox algorithm. As I understand it, the generalized box fold modifies reflections in the "BoxFold" step of the algorithm, whereas shape inversion modifies spherical inversion (and/or minimum radius) in the "SphereFold" step.

Both approaches share a common geometric concept of replacing one shape for another. This allowed me to translate and reuse part of your excellent generalized box fold code when I needed intersection calculations for the non-cube Platonic solids (tetrahedron, octahedron, icosahedron and dodecahedron). In my implementations I reference this reuse and include a link to the generalized box fold thread.

One clear difference between the results of the two approaches is that where the generalized box fold changes the overall shape of the fractal, shape inversion (usually) does not. Here's a few example images of the exterior of shape inversion Mandelboxen, showing that while the detailed structure does change, the overall shape remains a cube. Images are for standard spherical inversion, cube inversion, octahedral inversion and inversion using a linear blend between sphere and cube.

And the same set of shape inversions again: sphere, cube, octahedron, sphere/cube blend. But this time a face-on view zoomed into a corner, and with the Fragmentarium raytracer detail setting turned up higher. At this scale the differences compared to standard spherical inversion become clearer.

One direction I'd really like to explore is using both generalized box fold and shape inversion approaches in one modified Mandelbox algorithm. So that for example one could have the BoxFold use an octahedron for reflections while the SphereFold uses a rounded cube for inversions. Has anyone implemented the generalized box fold in Fragmentarium? I'm already having performance issues with the part I translated to reuse for shape intersections, so I'm not sure how practical this is in Fragmentarium. My GLSL code is extremely unoptimized though