Cation ordering in ABO3 perovskites adds to their chemical variety and can lead to properties such as ferrimagnetism and magnetoresistance in Sr2FeMoO6. Through high-pressure and high-temperature synthesis, a new type of “double double perovskite” structure has been discovered in the family MnRMnSbO6 (R=La, Pr, Nd, Sm). This tetragonal structure has a 1:1 order of cations on both A and B sites, with A-site Mn2+ and R3+ cations ordered in columns and Mn2+ and Sb5+ having rock salt order on the B sites. The MnRMnSbO6 double double perovskites are ferrimagnetic at low temperatures with additional spin-reorientation transitions. The ordering direction of ferrimagnetic Mn spins in MnNdMnSbO6 changes from parallel to [001] below TC=76 K to perpendicular below the reorientation transition at 42 K at which Nd moments also order. Smaller rare earths lead to conventional monoclinic double perovskites (MnR)MnSbO6 for Eu and Gd.