Spin-reorientation phase transitions that involve the rotation of a crystal's magnetization have been well characterized in distorted-perovskite oxides such as orthoferrites. In these systems spin reorientation occurs due to competing rare-earth and transition metal anisotropies coupled via f-d exchange. Here, we demonstrate an alternative paradigm for spin reorientation in distorted perovskites. We show that the R_{2}CuMnMn_{4}O_{12} (R=Y or Dy) triple A-site columnar-ordered quadruple perovskites have three ordered magnetic phases and up to two spin-reorientation phase transitions. Unlike the spin-reorientation phenomena in other distorted perovskites, these transitions are independent of rare-earth magnetism, but are instead driven by an instability towards antiferromagnetic spin canting likely originating in frustrated Heisenberg exchange interactions, and the competition between Dzyaloshinskii-Moriya and single-ion anisotropies.