  The purpose of this article is to shed new light on the combinatorial
structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main
focus is the set $\D$ of distinguished involutions in $W$, which was introduced
by Lusztig in one of his first papers on cells in affine Weyl groups. We
conjecture that the set $\D$ has a simple recursive structure and can be
enumerated algorithmically starting from the distinguished involutions of
finite Coxeter groups. Moreover, to each element of $\D$ we assign an
explicitly defined set of equivalence relations on $W$ that altogether
conjecturally determine the partition of $W$ into left (right) cells. We are
able to prove these conjectures only in a special case, but even from these
partial results we can deduce some interesting corollaries.
