Cohomology classes on moduli of curves from theta characteristics

Using Galois-Stiefel-Whitney classes of theta characteristics we show that over a totally real base field the moduli stack of smooth genus g curves and the moduli stack of principally polarized abelian varieties of dimension g have nontrivial cohomological invariants and étale cohomology classes in degree respectively 2^{g-2}, 2^{g-1} and 2^{g-1}. We also compute the pullback from the Brauer group of \mathcal{M}_3 to that of \mathcal{H}_3 over a general field of characteristic different from 2.