  Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an
even dimensional vector space equipped with a nondegenerate symmetric form. We
prove a Giambelli formula which expresses an arbitrary Schubert class in the
singular and quantum cohomology ring of X as a polynomial in certain special
Schubert classes. Our analysis reveals a surprising relation between the
Schubert calculus on even and odd orthogonal Grassmannians. We also study eta
polynomials, a family of polynomials defined using raising operators whose
algebra agrees with the Schubert calculus on X.
