In 3D, each topologically distinct polyhedron has a "canonical form" where all of its edges are tangent to the same sphere. The corresponding dual polyhedron can then be constructed by cutting a vertex figure off the midpoints of the original polyhedron's edges, and then building pyramids on the original faces so that they lie on the same plane as the vertex figure.

However, I'm not sure if this should even apply to 4D, because edges invert to faces instead, but dual uniform polychora with constant dichoral angles must exist, so how are they defined?