  The orbits of the orthogonal and symplectic groups on the flag variety are in
bijection, respectively, with the involutions and fixed-point-free involutions
in the symmetric group $S_n$. Wyser and Yong have described polynomial
representatives for the cohomology classes of the closures of these orbits,
which we denote as $\hat{\mathfrak{S}}_y$ (to be called involution Schubert
polynomials) and $\hat{\mathfrak{S}}^{\tt FPF}_y$ (to be called
fixed-point-free involution Schubert polynomials). Our main results are
explicit formulas decomposing the product of $\hat{\mathfrak{S}}_y$
(respectively, $\hat{\mathfrak{S}}^{\tt FPF}_y$) with any $y$-invariant linear
polynomial as a linear combination of other involution Schubert polynomials.
These identities serve as analogues of Lascoux and Sch\"utzenberger's
transition formula for Schubert polynomials, and lead to a self-contained
algebraic proof of the nontrivial equivalence of several definitions of
$\hat{\mathfrak{S}}_y$ and $\hat{\mathfrak{S}}^{\tt FPF}_y$ appearing in the
literature. Our formulas also imply combinatorial identities about involution
words, certain variations of reduced words for involutions in $S_n$. We
construct operators on involution words based on the Little map to prove these
identities bijectively. The proofs of our main theorems depend on some new
technical results, extending work of Incitti, about covering relations in the
Bruhat order of $S_n$ restricted to involutions.
