  We examine the relationship between the (double) Schubert polynomials of
Billey-Haiman and Ikeda-Mihalcea-Naruse and the (double) theta and eta
polynomials of Buch-Kresch-Tamvakis and Wilson from the perspective of Weyl
group invariants. We obtain generators for the kernel of the natural map from
the corresponding ring of Schubert polynomials to the (equivariant) cohomology
ring of symplectic and orthogonal flag manifolds.
