We consider differential Picard-Vessiot extensions as Hopf-bi-Galois extensions, an algebraic analogue of bitorsors, with respect to the coordinate rings of the intrinsic and special differential Galois groups. At one hand this gives rise to a new characterization of the intrinsic differential Galois group. At the other hand we obtain new Galois correspondences involving subalgebras of the Picard-Vessiot rings and certain objects related to the intrinsic and special Galois groups. This is joint work with Andreas Maurischat.