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A polyhedron is a graph 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 in 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 Vázsonyi problem. (Javier Bracho),doi:10.26493/1855-3974.2194.eab fatcat:hf4swwujwvhzjk6u5skdcpbqiu