This talk is about joint work with T. Chinburg, B. Morin and M.J. Taylor. Our aim is to establish comparison formulas between the Hasse-Witt invariants of a symmetric bundle over a scheme and the invariants of some of its twists. In particular we will consider a special kind of twist, which has been first studied by A. Fr\"ohlich for quadratic forms on fields. This arises from twisting the form by a cocycle obtained from an orthogonal representation of a group scheme. A simple important example of this twisting procedure is the trace form of an \´etale algebra, which is obtained by twisting the standard/sum of squares form by the orthogonal representation attached to the algebra. We will start by recalling the classical results of Fr\"ohlich and Serre for quadratic forms on fields. Then we will define the Hasse-Witt invariants one can associate to a symmetric bundle on a scheme and finally we will indicate some applications of our formulas to embedding problems.