  Involution Schubert polynomials represent cohomology classes of $K$-orbit
closures in the complete flag variety, where $K$ is the orthogonal or
symplectic group. We show they also represent $T$-equivariant cohomology
classes of subvarieties defined by upper-left rank conditions in the spaces of
symmetric or skew-symmetric matrices. This geometry implies that these
polynomials are positive combinations of monomials in the variables $x_i +
x_j$, and we give explicit formulas of this kind as sums over new objects
called involution pipe dreams. Our formulas are analogues of the
Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and
Miller's approach to matrix Schubert varieties, pipe dream formulas reflect
Gr\"obner degenerations of the ideals of those varieties, and we conjecturally
identify analogous degenerations in our setting.
