We prove an exact finite-volume symmetry formula for two-point functions in the periodic N-state superintegrable chiral Potts spin chain. We show that, for every chain length L and every simultaneous eigenvector of the Hamiltonian and the one-site translation operator, the correlations satisfy ⟨ Z₀ʳ Z_R^(† r)⟩^*=⟨ Z₀ʳ Z_(L-R)^(† r)⟩ for 1leqslant rleqslant N-1. Hence, whenever L is even, the midpoint correlation ⟨ Z₀ʳ Z_(L/2)^(† r)⟩ is real. Then we generalise the three-state chain case to arbitrary N and to every translation eigensector. This resolves a conjecture of Fabricius and McCoy.