The vertices of each cell of a simplicial complex are often assumed to have an ordering, for example in these notes, this exercise, and this blog post. From the ordering of the vertices of a simplex $\sigma$ and the ordering in a face $\tau$, you can calculate the incidence number $[\sigma : \tau]$ between them by evaluating the sign of a permutation. From the incidence numbers you can define boundary operators on chains and everything you need for homology. For example, in 2D Euclidean space this means that all the triangles go around counterclockwise. Many statements about ordering can be turned into statements about determinants of matrices formed from the vertices of each triangle.
The incidence numbers between cell and faces in CW complexes, on the other hand, seem to always be defined through the degree of the attaching map. There's no ordering assumed at all, but if you can define the incidence numbers then you can also define boundary operators and thus homology.
Is ordering a special property of simplicial complexes? For example, one could also define incidence numbers and ultimately calculate homology for more general spaces constructed from gluing together polyhedra with arbitrary numbers of faces and vertices. In every reference I've seen (for example this paper or Rourke and Sanderson, Introduction to Piecewise-Linear Topology) that does so, the incidence numbers are defined through the degrees of the attaching maps and there's no mention of ordering. Why is that? Can the incidence numbers be calculated through some ordering of the vertices of each cell and it's merely inconvenient computationally? Or is there some polyhedral complex for which the incidence numbers between a cell and its faces cannot be recovered from any ordering of the vertices?
My hunch is that ordering is meaningful only for simplices. The Herschel graph is both polyhedral and non-Hamiltonian, which seems to me to imply that there's no way at all to recover the incidence numbers from any ordering of the vertices. But I'm not great in algebraic topology so I might have misunderstood everything.