  We establish comparison results between the Hasse-Witt invariants w_t(E) of a
symmetric bundle E over a scheme and the invariants of one of its twists
E_{\alpha}. For general twists we describe the difference between w_t(E) and
w_t(E_{\alpha}) up to terms of degree 3. Next we consider a special kind of
twist, which has been studied by A. Fr\"ohlich. This arises from twisting by a
cocycle obtained from an orthogonal representation. We show how to explicitly
describe the twist for representations arising from very general tame actions.
This involves the ``square root of the inverse different'' which Serre,
Esnault, Kahn, Viehweg and ourselves had studied before. For torsors we show
that, in our geometric set-up, Jardine's generalization of Frohlich's formula
holds. The case of genuinely tamely ramified actions is geometrically more
involved and leads us to introduce a ramification invariant which generalises
in higher dimension the invariant introduced by Serre for curves.
