In this talk I will introduce a quantum chain generalizing the Zn clock model by keeping nearest-neighbour interaction and self-duality, but further requiring U(1) invariance. This model is integrable, and can be related to XXZ-type chains at their “roots of unity'' points, which present a number of striking features including a rich structure of degeneracies. Such a correspondence allows to revisit these features in an interesting way, in particular I will discuss the physics of the “exact complete strings'' underlying degeneracies in the Bethe ansatz language, and, in close connection with the latter, the presence of an underlying Onsager symmetry algebra.