  Suppose $X$ is a hyperelliptic curve of genus $g$ defined over an
algebraically closed field $k$ of characteristic $p=2$. We prove that the de
Rham cohomology of $X$ decomposes into pieces indexed by the branch points of
the hyperelliptic cover. This allows us to compute the isomorphism class of the
$2$-torsion group scheme $J_X[2]$ of the Jacobian of $X$ in terms of the
Ekedahl-Oort type. The interesting feature is that $J_X[2]$ depends only on
some discrete invariants of $X$, namely, on the ramification invariants
associated with the branch points. We give a complete classification of the
group schemes which occur as the $2$-torsion group schemes of Jacobians of
hyperelliptic $k$-curves of arbitrary genus.
