  We develop a formalism that allows a complete classification of
four-dimensional Z2XZ2 heterotic string models. Three generation models in a
sub-class of these compactifications are related to the existence of three
twisted sectors in Z2XZ2 orbifolds. In the work discussed here we classify the
sub-class of these models that produce spinorial representations from all three
twisted planes, and including symmetric shifts on the internal lattice. We show
that perturbative three generation models are not obtained solely with
symmetric shifts on complex tori, but necessitate the action of an asymmetric
shift. In a subclass of these models we show that their chiral content is
predetermined by the choice of the N=4 lattice. The implication of the results
and possible geometrical interpretation are briefly discussed.
