This paper considers the robust output regulation problem for parameterized families of periodic systems. To extend the solution of the output regulation problem to the periodic (or time-varying) setup, a classification of the immersion mappings based on various non-equivalent observability properties is derived. The connections between different canonical realizations of internal models that fully exploit such properties for robust and adaptive output regulation design in periodic systems are investigated. It is shown how non-minimal realizations of suitable periodic internal models are instrumental in achieving the possibility of performing adaptive redesign to deal with parameterized families of exosystem models. An important feature of the proposed solution is the fact that a persistence of excitation condition for the exogenous signals is not required for asymptotic regulation.