  According to an old result of Sch\"utzenberger, the involutions in a given
two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper,
we study possible generalisations of this property to other types of Coxeter
groups. We show that Sch\"utzenberger's result is a special case of a general
result on "smooth" two-sided cells. Furthermore, we consider Kottwitz'
conjecture concerning the intersections of conjugacy classes of involutions
with the left cells in a finite Coxeter group. Our methods lead to a proof of
this conjecture for classical types; combined with previous work, this leaves
type $E_8$ as the only remaining open case.
