  Pattern-equivariant (PE) cohomology is a well-established tool with which to
interpret the \v{C}ech cohomology groups of a tiling space in a highly
geometric way. In this paper we consider homology groups of PE infinite chains.
We establish Poincar\'{e} duality between the PE cohomology and PE homology.
The Penrose kite and dart tilings are taken as our central running example, we
show how through this formalism one may give highly approachable geometric
descriptions of the generators of the \v{C}ech cohomology of their tiling
space. These invariants are also considered in the context of rotational
symmetry. Poincar\'{e} duality fails over integer coefficients for the `ePE
homology groups' based upon chains which are PE with respect to
orientation-preserving Euclidean motions between patches. As a result we
construct a new invariant, which is of relevance to the cohomology of
rotational tiling spaces. We present an efficient method of computation of the
PE and ePE (co)homology groups for hierarchical tilings.
