  We extend Moonen's definition of Ekedahl-Oort types of smooth curves in terms
of Hasse-Witt triples to all stable curves and show that it matches Ekedahl and
van der Geer's definition of Ekedahl-Oort types of their generalized Jacobians
as semi-abelian varieties. Using this intrinsic insight, we can compute the
dimensions of certain Ekedahl-Oort loci of curves and generalize some
previously known results about the dimensions of the $p$-rank and $a$-number
loci of curves.
