  We characterize commutative idempotent involutive residuated lattices as
disjoint unions of Boolean algebras arranged over a distributive lattice. We
use this description to introduce a new construction, called gluing, that
allows us to build new members of this variety from other ones. In particular,
all finite members can be constructed in this way from Boolean algebras.
Finally, we apply our construction to prove that the fusion reduct of any
finite member is a distributive semilattice, and to show that this variety is
not locally finite.
