  This article is a follow up of the previous article of the authors on the
analytic surgery of eta- and rho-invariants. We investigate in detail the
(Atiyah-Patodi-Singer)-rho-invariant for manifolds with boundary. First we
generalize the cut-and-paste formula to arbitrary boundary conditions. A priori
the rho-invariant is an invariant of the Riemannian structure and a
representation of the fundamental group. We show, however, that the dependence
on the metric is only very mild: it is independent of the metric in the
interior and the dependence on the metric on the boundary is only up to its
pseudo--isotopy class. Furthermore, we show that this cannot be improved: we
give explicit examples and a theoretical argument that different metrics on the
boundary in general give rise to different rho-invariants. Theoretically, this
follows from an interpretation of the exponentiated rho-invariant as a
covariantly constant section of a determinant bundle over a certain moduli
space of flat connections and Riemannian metrics on the boundary. Finally we
extend to manifolds with boundary the results of Farber-Levine-Weinberger
concerning the homotopy invariance of the rho-invariant and spectral flow of
the odd signature operator.
