  An orthogonal bundle over a curve has an isotropic Segre invariant determined
by the maximal degree of a Lagrangian subbundle. This invariant, and the
induced stratifications on moduli spaces of orthogonal bundles, were studied
for bundles of even rank in [4]. In this paper, we obtain analogous results for
bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre
invariant. We show the irreducibility of the induced strata on the moduli
spaces of orthogonal bundles of odd rank, and compute their dimensions. As a
key ingredient of the proofs, we study the correspondence between Lagrangian
subbundles of orthogonal bundles of even and odd rank.
