  The action of a Coxeter group $W$ on the set of left cosets of a standard
parabolic subgroup deforms to define a module $\mathcal{M}^J$ of the group's
Iwahori-Hecke algebra $\mathcal{H}$ with a particularly simple form. Rains and
Vazirani have introduced the notion of a quasiparabolic set to characterize
$W$-sets for which analogous deformations exist; a motivating example is the
conjugacy class of fixed point free involutions in the symmetric group. Deodhar
has shown that the module $\mathcal{M}^J$ possesses a certain antilinear
involution, called the bar operator, and a certain basis invariant under this
involution, which generalizes the Kazhdan-Lusztig basis of $\mathcal{H}$. The
well-known significance of this basis in representation theory makes it natural
to seek to extend Deodhar's results to the quasiparabolic setting. In general,
the obstruction to finding such an extension is the existence of an appropriate
quasiparabolic analogue of the "bar operator." In this paper, we consider the
most natural definition of a quasiparabolic bar operator, and develop a theory
of "quasiparabolic Kazhdan-Lusztig bases" under the hypothesis that such a bar
operator exists. Giving content to this theory, we prove that a bar operator in
the desired sense does exist for quasiparabolic $W$-sets given by twisted
conjugacy classes of twisted involutions. Finally, we prove several results
classifying the quasiparabolic conjugacy classes in a Coxeter group.
