A closed, smooth, orientable 4-manifold X with order two fundamental group is determined up to homeomorphism by its intersection form and its w2-type, i.e. whether it and its double cover admit spin structures. There are three possible options for the w2-type: (1) X and its double cover are both non-spin, (2) X and its double cover are both spin, and (3) X is non-spin but its double cover is spin. In this talk, we’ll review fundamental properties of spin structures, from which we’ll derive tools to determine the w2-type of a given manifold. We’ll use this machinery to construct irreducible 4-manifolds for each possible w2-type. This is joint work with Mihail Arabadji.