  We construct eta- and rho-invariants for Dirac operators, on the universal
covering of a closed manifold, that are invariant under the projective action
associated to a 2-cocycle of the fundamental group. We prove an
Atiyah-Patodi-Singer index theorem in this setting, as well as its higher
generalization. Applications concern the classification of positive scalar
curvature metrics on closed spin manifolds. We also investigate the properties
of these twisted invariants for the signature operator and the relation to the
higher invariants.
