  The Ekedahl-Oort type is a combinatorial invariant of a principally polarized
abelian variety $A$ defined over an algebraically closed field of
characteristic $p > 0$. It characterizes the $p$-torsion group scheme of $A$ up
to isomorphism. Equivalently, it characterizes (the mod $p$ reduction of) the
Dieudonn\'e module of $A$ or the de Rham cohomology of $A$ as modules under the
Frobenius and Vershiebung operators.
  There are very few results about which Ekedahl-Oort types occur for Jacobians
of curves. In this paper, we consider the class of Hermitian curves, indexed by
a prime power $q=p^n$, which are supersingular curves well-known for their
exceptional arithmetic properties. We determine the Ekedahl-Oort types of the
Jacobians of all Hermitian curves. An interesting feature is that their
indecomposable factors are determined by the orbits of the
multiplication-by-two map on ${\mathbb Z}/(2^n+1)$, and thus do not depend on
$p$. This yields applications about the decomposition of the Jacobians of
Hermitian curves up to isomorphism.
