  We study the relationship between two notions of pattern avoidance for
involutions in the symmetric group and their restriction to fixed-point-free
involutions. The first is classical, while the second appears in the geometry
of certain spherical varieties and generalizes the notion of pattern avoidance
for perfect matchings studied by Jel\'inek. The first notion can always be
expressed in terms of the second, and we give an effective algorithm to do so.
We also give partial results characterizing the families of involutions where
the converse holds. As a consequence, we prove two conjectures of McGovern
characterizing (rational) smoothness of certain varieties. We also give new
enumerative results, and conclude by proposing several lines of inquiry that
extend our current work.
