Some insight might be gained by considering barycentric coordinates for the dual diplo-simplex. In two dimensions, we have the coordinates of the hexagon as the six permuations of (0, 1/3, 2/3). In three dimensions, the vertices of the octahedron are the six permutations of (0, 0, 1/2, 1/2).
Let d - 1 represent the number of dimensions. This simplifies some formatting, since barycentric coordinates will then have d terms.
In general, for even d, we have for vertices all permutations of (0, ..., 0, 2/d, ...,2/d), where there are d/2 copies each of 0 and 2/d. When d is odd, the vertices are all permutations of (0, ..., 0, 1/d, 2/d, ..., 2/d), where 0 and 2/d each occur (d-1)/2 times. (This was suggested by some work with Mathematica; the number of vertices works out, though I don't have a rigorous proof at the moment.)
So in four dimensions, the vertices are permutations of (0, 0, 1/5, 2/5, 2/5). This polytope has 10 truncated tetrahedra (the Archimedean truncation) as cells; the dual simplex truncates the original simplex in a nice way.
In five dimensions, we have permutations of (0, 0, 0, 1/3, 1/3, 1/3), so that the faces are 12 completely truncated 4-simplices (in the sense that the octahedron is the complete truncation of the tetrahedron).
The advantage of using barycentric coordinates is that the vertices are easy to describe, the symmetry is evident, and so the combinatorics are relatively easy to discern.
Was there a specific context in which this polytope arose? Are there any specific combinatorial properties of this polytope which are needed?