I am considering the following transformation on two-dimensional cell-complexes—aka polyhedral surfaces. (It can be defined in higher dimension, but I only care about dimension two.) Since it is obvious, I suppose it is well-known, and I would like to know what its name is.

Assuming no boundary,

- each edge is doubled, i.e. replaced by a pair of edges,
- each vertex is replaced by $d$ vertices, where $d$ is the vertex’ degree
- extra faces are added between pairs of edges and at each blown vertex.

Here's an example with the tetrahedron (where the outside represents the fourth face):

Note that this transformation is not exactly a subdivision since some of the previous elements (edges and vertices) are not preserved.
It may pictured and realized for a convex polyhedron $K$ by a *scraping* operation, slicing by a plane at each vertex and along each edge. Or, more mathematically, by interpolating between $K$ and its polar $K^\circ$.

Has anybody met this operation?