Abstract: The eta invariant of the spin^c Dirac operator measures the asymmetry of its spectrum about zero. In this talk, I will discuss the eta invariant of Seifert rational homology 3-spheres and how it controls the grading shift in Floer homology. When the Seifert base is non-orientable, it turns out that the eta invariant completelly determines the Floer homotopy type of the manifold and its computation involves a contribution by an orbifold pin^c eta invariant.