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An element of a Coxeter group $W$ is fully commutative if any two of itsreduced decompositions are related by a series of transpositions of adjacentcommuting generators. In the present work, we focus on fully commutativeinvolutions, which are characterized in terms of Viennot's heaps. By encodingthe latter by Dyck-type lattice walks, we enumerate fully commutativeinvolutions according to their length, for all classical finite and affineCoxeter groups. In the finite cases, we also find explicit expressions fortheir generating functions with respect to the major index. Finally in affinetype $A$, we connect our results to Fan--Green's cell structure of thecorresponding Temperley--Lieb algebra.

Combinatorics of fully commutative involutions in classical Coxeter groups