Traditionally, most studies of quantum many-body systems have been mainly concerned with properties of the states of low-lying energy. Recently, however, fascinating features of the full energy spectrum have been uncovered. Among these are eigenstate phase transitions, where sharp transitions occur not only in the ground state, but in all the states. I describe a simple example of such, a transition for a strong zero mode in the XYZ spin chain. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in parafermionic systems and strongly disordered many-body localized phases. My proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure – despite being a rather elaborate operator, it squares to the identity.