  In this paper, we provide constructions to enumerate large numbers of
CI-liaison classes. To this end, we introduce a liaison invariant and prove
several results concerning it, notably that it commutes with hypersurface
sections. This theory is applied to the CI-liaison classes of ruled joins of
projective schemes, yielding strong obstructions for such joins to lie in the
same liaison class. A second construction arises from the actions of
automorphisms on liaison classes, allowing the enumeration of many liaison
classes of perfect ideals of codimension at least three.
