The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point of view). There are probably more purely extensional sets, with similar properties. I just wanted to pin down one concrete example. (From an intensional point of view however, some of these sets might depend strongly on the underlying set universe. Take the natural numbers $\omega$ as an example.)
Do there exists a dual thing for classes, i.e. purely intensional classes? These would be proper classes described intensionally, that are quite independent of the underlying set universe (at least if we look at them from an intensional point of view). A concrete example could be the class of all sets (i.e. the dual notion of the empty set). But what about the class of all sets with two elements? Or the class of all abelian groups? Or the class of all sets different from the sets in a given hereditarily finite set?