Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the category of (oriented and abstract) simplicial complexes. An object in this category consists of a set together with a collection of its subsets, called simplices, each equipped with a total order. The collection of simplices contains all singletons, is close under taking subsets, and the order of a simplex agrees with the one induced from any simplex containing it.

Is there an interesting combinatorially defined subcategory of cubical sets? Maybe one analogue to simplicial complexes?

Cubical Homotopy Theory: A Beginning. I doesn't quite give the definition that you want, but maybe it comes close. – Karol Szumiło yesterday