Neighbourhood structures are the standard semantic tool used to reason about
non-normal modal logics. The logic of all neighbourhood models is called
classical modal logic. In coalgebraic terms, a neighbourhood frame is a
coalgebra for the contravariant powerset functor composed with itself, denoted
by 2². We use this coalgebraic modelling to derive notions of
equivalence between neighbourhood structures. 2²-bisimilarity and
behavioural equivalence are well known coalgebraic concepts, and they are
distinct, since 2² does not preserve weak pullbacks. We introduce a
third, intermediate notion whose witnessing relations we call precocongruences
(based on pushouts). We give back-and-forth style characterisations for
2²-bisimulations and precocongruences, we show that on a single
coalgebra, precocongruences capture behavioural equivalence, and that between
neighbourhood structures, precocongruences are a better approximation of
behavioural equivalence than 2²-bisimulations. We also introduce a
notion of modal saturation for neighbourhood models, and investigate its
relationship with definability and image-finiteness. We prove a Hennessy-Milner
theorem for modally saturated and for image-finite neighbourhood models. Our
main results are an analogue of Van Benthem's characterisation theorem and a
model-theoretic proof of Craig interpolation for classical modal logic.