  We give explicit formulas for torus-equivariant fundamental classes of closed
$K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups
$SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$.
We describe parametrizations of each orbit set and the combinatorics of its
weak order, allowing us to compute formulas for the equivariant classes of all
remaining orbit closures using divided difference operators. In each of the
cases in type A, we realize the $K$-orbit closures as universal degeneracy loci
of a certain type, involving a vector bundle $V$ over a scheme $X$ equipped
with a flag of subbundles and a further structure determined by $K$. We
describe how our equivariant formulas can be interpreted as formulas for such
loci in the Chern classes of the various bundles on $X$.
