In this talk I will discuss the Feynman and anti-Feynman inverses for wave operators on certain Lorentzian (and more generally pseudo-Riemannian) manifolds; these are two inverses which from a microlocal analysis perspective are more natural than the standard causal (advanced/retarded) ones. For instance, for the spectral family of the wave operator, these are the natural inverses when the spectral parameter is non-real. Indeed, I will explain that these connect to the self-adjointness of the wave operator, and the positivity properties that follow. (Based in part on joint work with Dang, Gell-Redman, Haber and Wrochna.)