  We give the formula for multiplying a Schubert class on an odd orthogonal or
symplectic flag manifold by a special Schubert class pulled back from a
Grassmannian of maximal isotropic subspaces. This is also the formula for
multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the
Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$).
Geometric constructions and intermediate results allow us to ultimately deduce
this from formulas for the classical flag manifold. These intermediate results
are concerned with the Bruhat order of the Coxeter group ${\mathcal B}_\infty$,
identities of the structure constants for the Schubert basis of cohomology, and
intersections of Schubert varieties. We show these identities follow from the
Pieri-type formula, except some `hidden symmetries' of the structure constants.
Our analysis leads to a new partial order on the Coxeter group ${\mathcal
B}_\infty$ and formulas for many of these structure constants.
