A class of quadruple systems called dihedral quadruple systems (DQSs), which lie between Mendelsohn and Steiner quadruple systems, is defined and considered. A ternary quasigroup which is invariant under conjugation by every permutation from $D_4$, the dihedral group, is called dihedral. It is proved that DQSs are equivalent to generalized idempotent dihedral ternary quasigroups. Some properties of such ternary quasigroups and DQSs are described and their spectrum determined. It is proved that a DQS of order $v$ exists if and only if $v\equiv0(\mod 2)$, $v\geq 4$.