  With a scalar field non-minimally coupled to curvature, the underlying
geometry and variational principle of gravity - metric or Palatini - becomes
important and makes a difference, as the field dynamics and observational
predictions generally depend on this choice. In the present paper we describe a
classification principle which encompasses both metric and Palatini models of
inflation, employing the fact that inflationary observables can be neatly
expressed in terms of certain quantities which remain invariant under conformal
transformations and scalar field redefinitions. This allows us to elucidate the
specific conditions when a model yields equivalent phenomenology in the metric
and Palatini formalisms, and also to outline a method how to systematically
construct different models in both formulations that produce the same
observables.
