  There is a natural analogue of weak Bruhat order on the involutions in any
Coxeter group. The saturated chains of intervals in this order correspond to
reduced words for a certain set of group elements called atoms. Brion gives a
general formula for the cohomology class of a $K$-orbit closure in an arbitrary
flag variety, where $K$ is a symmetric subgroup of a complex algebraic group.
In type A, the terms in this formula are indexed by atoms for permutations. We
study the combinatorics of atoms for involutions in the group of signed
permutations. In particular, we give a compact description of the atom set for
any signed involution and endow it with the structure of a graded poset. Our
main result, as an application, is to identify explicitly the terms in Brion's
cohomology formula in types B and C. These descriptions apply to all $K$-orbits
in these types and are the first of their kind outside of type A.
