First we prove that the class $C_{I}$ of centrally symmetric convex polyhedra with regular polygonal faces consists of 4 of the 5 Platonic, 9 of the 13 Archimedean, 13 of the 92 Johnson solids and two infinite families of $2n$-prisms and $(2n+1)$-antiprisms. Then we show how the presented maps of their halves (obtained by identification of all pairs of antipodal points) in the projective plane can be used for obtaining their flag graphs and symmetry-type graphs. Finally, we study some linear dependence relations between polyhedra of the class $C_{I}$.