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Involution words are variations of reduced words for involutions in Coxetergroups, first studied under the name of "admissible sequences" by Richardsonand Springer. They are maximal chains in Richardson and Springer's weak orderon involutions. This article is the first in a series of papers on involutionwords, and focuses on their enumerative properties. We define involutionanalogues of several objects associated to permutations, including Rothediagrams, the essential set, Schubert polynomials, and Stanley symmetricfunctions. These definitions have geometric interpretations for certainintervals in the weak order on involutions. In particular, our definition of"involution Schubert polynomials" can be viewed as a Billey-Jockusch-Stanleytype formula for cohomology class representatives of $\mathrm{O}_n$- and$\mathrm{Sp}_{2n}$-orbit closures in the flag variety, defined inductively inrecent work of Wyser and Yong. As a special case of a more general theorem, weshow that the involution Stanley symmetric function for the longest element ofa finite symmetric group is a product of staircase-shaped Schur functions. Thisimplies that the number of involution words for the longest element of a finitesymmetric group is equal to the dimension of a certain irreduciblerepresentation of a Weyl group of type $B$.

Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures