It is shown, by density-functional theory calculations, that gold clusters on the MgO(001) surface prefer cage structures in the size range between 23 and 42 atoms. These structures belong to a new structural family, the open pyramidal hollow cages, which has no counterpart in gas-phase clusters. These cages are possible because of the peculiar features of the Au-Au and Au-MgO interactions, which include strong many-body and directional effects. These effects reinforce the tendency of Au to produce cage structures with respect to the gas phase.