I'll define a new characteristic class for two types of objects associated to the group GO_n of orthogonal similitudes: line bundle-valued quadratic forms over algebraic varieties, and so-called "essentially self-dual" Galois representations. This characteristic class "interpolates" between the classical 2nd Stiefel-Whitney (or Hasse-Witt) class and the 1st Chern class modulo 2. I'll give some applications to the classification of line bundle-valued quadratic forms of low rank and to some index 4 embedding problems in Galois theory.