The symmetry of tetracoordinated copper complexes, especially of CuCl4(2-), is analyzed in terms of quantitative continuous symmetry. These complexes acquire structures that span from tetrahedral to planar, with all gradual variations in between, a property that is evaluated here in terms of their degrees of tetrahedricity (S(Td)) and square planarity (S(D4h)). It was found that out of the large arsenal of geometry-allowed tetrahedral structures, each of which is characterized by specific S(Td) and S(D4h) pair values, copper complexes concentrate on an extremely well-defined correlation line linking these two symmetry-content measures; that is, only very specific pairs of S(Td) and S(D4h) values that dictate each other are allowed. Furthermore, it was found that of the various routes that can lead from a tetrahedron to a planar square, the mode known as spread fits exactly in its symmetry S(Td)/S(D4h) characteristics the observed symmetry behavior of the copper complexes. Interestingly, the spread mode reflects also the (nearly) minimal possible values of S(Td)/S(D4h) pairs, namely, the minimal symmetry-distortive route. Hints that this symmetry correlation line reflects universal features that go beyond copper are provided by the symmetry-content analysis of Ni and Pt complexes, of various Zn complexes within a metalloprotein and even of CC4 fragments, all of which obey the same line. A new potential-energy surface is introduced, the axes of which are S(Td), S(D4h), and the energy. Calculating this surface for CuCl4(2-) reveals an important result: The minimal molecular symmetry distortive spread correlation line coincides with the (only) energy valley of the map. Symmetry and energy are intimately related in their drive to minimal values.