The classical cube of non-normal modal logics is considered, and an alternative neighbourhood semantics is given in which worlds are equipped with sets of pairs of neighbourhoods. The intuition is that the two neighbourhoods of a pair provide independent positive and negative evidence (or support) for a formula. This bi-neighbourhood semantics is significant in particular for logics without the monotonicity property. It is shown that this semantics characterises the cube of non-normal modal logics and that there is a mutual correspondence between models in the standard and in the bi-neighbourhood semantics. On the basis of this alternative semantics, labelled sequent calculi are developed for all the logics of the classical cube. The calculi thus obtained are fully modular and have good structural properties, first of all, syntactic cut elimination. Moreover, they provide a decision procedure and an easy countermodel extraction, both in the bi-neighbourhood and in the standard semantics.