The phase diagrams of cuprate superconductors and of QCD at non-zero baryon chemical potential are qualitatively similar. The Néel phase of the cuprates corresponds to the chirally broken phase of QCD, and the high-temperature superconducting phase corresponds to the color superconducting phase. In the SO(5) theory for the cuprates the SO(3) s spin rotational symmetry and the U(1) em gauge symmetry of electromagnetism are dynamically unified. This suggests that the SU(2) L ⊗ SU(2) R ⊗ U(1) B chiral symmetry of QCD and the SU(3) c color gauge symmetry may get unified to SO(10). Dynamical enhancement of symmetry from SO(2) s ⊗ Z(2) to SO(3) s is known to occur in anisotropic antiferromagnets. In these systems the staggered magnetization flops from an easy 3-axis into the 12-plane at a critical value of the external magnetic field. Similarly, the phase transitions in the SO(5) and SO(10) models are flop transitions of a "superspin". Despite this fact, a renormalization group flow analysis in 4 — ɛ dimensions indicates that a point with full SO(5) or SO(10) symmetry exists neither in the cuprates nor in QCD.