  This article studies a few new insertion algorithms that associate pairs of
shifted tableaux to finite integer sequences in which certain terms may be
primed. When primes are ignored in the input word these algorithms reduce to
known correspondences, namely, a shifted form of Edelman-Greene insertion,
Sagan-Worley insertion, and Haiman's mixed shifted insertion. The latter maps
have the property that when the input word varies such that one of the output
tableaux is fixed, the other output tableau ranges over all (semi)standard
tableau of a given shape with no primed entries on the diagonal. Our algorithms
have the same feature, but now with primes allowed on the diagonal. One
application of this is to give another Littlewood-Richardson rule for products
of Schur $Q$-functions. It is hoped that there will exist "set-valued"
generalizations of our bijections that can be used similarly to understand
products of $K$-theoretic Schur $Q$-functions. We also discuss how Hansson and
Hultman's analogue of Matsumoto's theorem for involution words extends to the
primed setting.
