  The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits
of the analogues of Schubert polynomials for the orbits of the orthogonal group
in the flag variety. These symmetric functions are also generating functions
for involution words, and are indexed by the involutions in the symmetric
group. By construction each $\hat{F}_y$ is a sum of Stanley symmetric functions
and therefore Schur positive. We prove the stronger fact that these power
series are Schur $P$-positive. We give an algorithm to efficiently compute the
decomposition of $\hat{F}_y$ into Schur $P$-summands, and prove that this
decomposition is triangular with respect to the dominance order on partitions.
As an application, we derive pattern avoidance conditions which characterize
the involution Stanley symmetric functions which are equal to Schur
$P$-functions. We deduce as a corollary that the involution Stanley symmetric
function of the reverse permutation is a Schur $P$-function indexed by a
shifted staircase shape. These results lead to alternate proofs of theorems of
Ardila-Serrano and DeWitt on skew Schur functions which are Schur
$P$-functions. We also prove new Pfaffian formulas for certain related
involution Schubert polynomials.
