It is known that strongly involutive polyhedra are closely related to self-dual maps where the antipodal function acts as duality isomorphism. Such a family of polyhedra appears in different combinatorial, topological and geometric contexts, and is thus attractive to be studied. In this note, we determine the 10 antipodal pairings among the classification of the 24 self-dual pairings $Dual(G)\rhd Aut(G)$ of self-dual maps G. We also present the orbifold associated to each antipodal pairing and describe explicitly the corresponding fundamental regions. We finally explain how to construct two infinite families of strongly involutive polyhedra (one of them new) by using their doodles and the action of the corresponding orbifolds.