  In this paper we study how certain symmetries of convex bodies affect their
geometric properties. In particular, we consider the impact of symmetries
generated by the block diagonal subgroup of orthogonal transformations,
generalizing complex and quaternionic convex bodies. We conduct a systematic
study of sections of bodies with symmetries of this type, with the emphasis on
problems of the Busemann-Petty type and hyperplane inequalities. The main role
belongs to the class of intersection bodies with symmetries.
