Towards the Classification of Z2×Z2 Fermionic Models

We develop a formalism that allows a complete classification of four-dimensional Z₂×Z₂ heterotic string models. Three generation models in a sub-class of these compactifications are related to the existence of three twisted sectors in Z₂×Z₂ 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.