In this substantial paper the author revisits the ordered groupoid approach to inverse and regular semigroups with a topological slant. Specifically it is proved that any ordered groupoid can be realized as the fundamental ordered groupoid of an ordered 2-complex. It is shown how to construct an ordered groupoid presentation from any inverse semigroup presentation. The so-called Schützenberger complex of an inverse semigroup presentation is introduced and it plays a role in the theory similar to that played by the Cayley complex in group theory. A new proof that a maximal subgroup of a full amalgam of regular semigroups has a certain graph of groups composition allows proof and extension of a range of standard results on strong embeddability of regular semigroup amalgams. Certain results in the category of inverse semigroups and pre-homomorphisms are likewise obtained.