(Top) Lattice structure of the iron-based superconductors. Note that the atomic positions are symmetric under a rotation by 90 degrees in the ab plane, followed by a reflection c→-c. This is the basic operation of the S4 symmetry. (Bottom) The gauge transformation consists of multiplying by a minus sign the electronic creation and annihilation operators on the orange iron sites. The transformation changes the pairing symmetry from s wave to d wave, as shown.