Two point symmetry rules are proven, specifying constraints on the extent of deformations of nuclear configurations which preserve the identity of chemical species. These rules formalize some of the relations between nuclear point symmetry and the gradient field within relaxed cross sections of potential energy hypersurfaces. Relaxed cross sections have lower dimension than the potential energy hypersurface; consequently, their generation by relaxation algorithms is simpler than the computation of domains of a hypersurface of high dimensions. These rules have utility in the search for critical points within relaxed cross sections, relying on the easily calculable point symmetry properties of nuclear arrangements.