  A polyhedron is a graph $G$ which is simple, planar and 3-connected. In this
note, we classify the family of strongly involutive self-dual polyhedra. The
latter is done by using a well-known result due to Tutte characterizing
3-connected graphs. We also show that this special class of polyhedra
self-duality behaves topologically as the antipodal mapping. These self-dual
polyhedra are related with several problems in convex and discrete geometry
including the V\'azsonyi problem.
