  We introduce shifted analogues of key polynomials related to symplectic and
orthogonal orbit closures in the complete flag variety. Our definitions are
given by applying isobaric divided difference operators to the analogues of
Schubert polynomials for orbit closures that correspond to dominant
involutions. We show that our shifted key polynomials are linear combinations
of key polynomials with nonnegative integer coefficients. We also prove that
they are partial versions of the classical Schur $P$- and $Q$-polynomials.
Finally, we examine $K$-theoretic generalizations of these functions, which
give shifted forms of Lascoux polynomials. In the symplectic case, these
generalizations are partial versions of the $GP$-polynomials introduced by
Ikeda and Naruse. Besides developing basic properties, we identify a number of
conjectures and open problems.
