Molecular ground states are found to have an approximate symmetry related to equally spaced inflection points from d(j)V(R)/dR(j) = 0. Morse, Kratzer-Coulomb, Rydberg, (n + l,n), exp-exp, and cubic-anharmonic potentials turn out to have exact equal spacing of all inflection points out to dissociation. Equal spacing near equilibrium is consistent with the rule (R-0+R-2)/2=R-e, connecting the hard-sphere radius and the point of maximum attractive bonding force to the equilibrium bond length. In theoretical and experimental molecular curves, the rule tends to be exact at high reduced force constant k(e), with symmetry breaking over k(e)=4-81 related to covalent, ionic, and van der Waals bonding character. Scaling preserves spacing symmetry, and maps two-term potentials into a universal exp-exp limit, including the (2n,n) potential into the Morse potential. Scaled spacing parameters for different molecules are nearly constant. Anharmonic shape parameters for "tilt" and "width" of the well are linked to empirical correlations of Dunham constants [J. L. Graves and R. G. Parr, Phys. Rev. A 31, 1 (1985)], and RKR analysis suggests correlations induced by equal-spacing constraints. The inflection structure is linked to threshold singularities in the inverse Born-Oppenheimer potential R(V), which predicts the (2n, n) potential as a first approximation.