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We establish comparison results between the Hasse–Witt invariants wt (E) of a symmetric bundle E over a scheme and the invariants of one of its twists Ea. For general twists we describe the difference between wt (E) and wt (Ea) up to terms of degree 3. Next we consider a special kind of twist, which has been studied by A. Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. A simple important example of this twisting procedure is the bilinear trace form of an étale algebra, which is obtained by twisting the standard/sum-of-squares form by the orthogonal representation attached to the algebra. 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, Jardines generalisation of Fröhlichs formula holds. Namely let (X, G) be a torsor with quotient Y, let E be a symmetric bundle over Y, let ?: G ? O(E) be an orthogonal representation and let E?, X be the corresponding twist of E. Then we verify up to degree 3 that the formula wt (E?, X) Spt (?) = wt (E) wt (?) holds. Here Spt (?) and wt (?) are respectively the spinor invariant and the Stiefel–Whitney class of ?. The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification, which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by Serre for curves.The comparison result in the tamely ramified case proceeds by reduction to the case of a torsor. The reduction is carried out by means of a partial normalisation procedure, which we had introduced in a previous paper. An important lemma of Esnault, Kahn and Viehweg allows us to express the difference between the invariants of bundles before and after the normalisation procedure in terms of Chern classes of certain sub-bundles. As noted elsewhere, this result can be best understood in the context of symmetric complexes and their invariants. Our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours. [ABSTRACT FROM AUTHOR]