  When the genus $g$ is even, we extend the computation of mod 2 cohomological
invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an
explicit functorial description of the invariants and we completely describe
their multiplicative structure. In the Appendix, we show that the cohomological
invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and
use our methods to give a very short proof of a result by Cornalba on the
Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it
to positive characteristic
