There is a beautiful Physics Viewpoint Untangling the Orbitals in Iron-Based Superconductors by Daniel Podolsky.
It describes a Phys. Rev. X paper by Jiangping Hu and Ningning Hao.
They show that a key to understanding the electronic structure of the iron pnictide and chalcogenide superconductors is the S4 symmetry of the lattice of Fe ions.
They are able to explain why the iron chalcogenides do not have holelike Fermi surface pockets whereas the iron pnictides do.
Furthermore, by means of a simple gauge transformation associated with the S4 symmetry they map the s-wave superconducting order parameter to the d-wave order parameter associated with the C_4v symmetry of the Cu ions in the cuprates.