  Higgs bundles over a closed orientable surface can be defined for any real
reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe
a rigidity phenomenon encountered in the case of maximal Toledo invariant.
Using this and Morse theory in the moduli space of Higgs bundles, we show that
the moduli space is connected in this maximal Toledo case. The Morse theory
also allows us to show connectedness when the Toledo invariant is zero. The
correspondence between Higgs bundles and surface group representations thus
allows us to count the connected components with zero and maximal Toledo
invariant in the moduli space of representations of the fundamental group of
the surface in SO*(2n).
