This Wikipedia page states that "zero modes appear whenever a physical system possesses a certain symmetry," and gives the example of a ring of beads connected by springs having a zero mode associated with rigidly rotating the whole system. It's easy to think of other examples too: most systems of beads connected by springs have zero modes that arise from translating the whole system (i.e. translational symmetry), for example.

Consider a system of four beads, connected by springs in a square at equilibrium. This system has 8 degrees of freedom, so it has 8 modes. The four nonzero modes come from either both the vertical or both the horizontal springs oscillating either in sync or 180 degrees out of sync. Two of the zero modes come from translation of the whole system, and the corresponding symmetry is obvious. One of the zero modes comes from rotating the whole system, and again, the corresponding symmetry is obvious.

However, the final zero mode comes from bringing two of the opposite corners together while pushing the other opposite two corners apart (i.e. squishing the square into a rhombus). So my question is this: what symmetry of the system does this deformation correspond to?