We develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

Formulas for the derived Witt groups of projective bundles are obtained. We deduce them from general properties of Witt theory, with the help of twisted Thom isomorphisms, avoiding explicit use of triangulated categories. Witt groups of completely split quadrics are also considered.

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.