  Let M be a compact smoothly stratified pseudomanifold with boundary,
satisfying the Witt assumption. In this paper we introduce the de Rham
signature and the Hodge signature of M, and prove their equality. Next,
building also on recent work of Albin and Gell-Redman, we extend the
Atiyah-Patodi-Singer index theory established in our previous work under the
hypothesis that M has stratification depth 1 to the general case, establishing
in particular a signature formula on Witt spaces with boundary.
  In a parallel way we also pass to the case of a Galois covering M' of M with
Galois group Gamma. Employing von Neumann algebras we introduce the de Rham
Gamma-signature and the Hodge Gamma-signature and prove their equality, thus
extending to Witt spaces a result proved by Lueck and Schick in the smooth
case. Finally, extending work of Vaillant in the smooth case, we establish a
formula for the Hodge Gamma-signature. As a consequence we deduce the
fundamental result that equates the Cheeger-Gromov rho-invariant of the
boundary of M' with the difference of the signatures of M and M'.
  We end the paper with two geometric applications of our results.
