A 4-cycle algebra is a finite-dimensional digraph algebra (CSL algebra) whose reduced digraph is a 4-cycle. A rigid embedding between such algebras is a direct sum of certain nondegenerate multiplicity one star-extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systems A of 4-cycle algebras with rigid embeddings. The critical invariant is a binary relation in K 0 A ร H 1 A, generalising the scale of the K 0 group, called the joint scale. The joint scale encapsulates other invariants and compatibility conditions of regular isomorphism. These include the scale of H 1 A, the scale of H 0 Aร H 1 A, sign compatibility, congruence compatibility and H 0 H 1 coupling classes. These invariants are also important for lifting K 0 รH 1 isomorphisms to algebra isomorphisms; we resolve this lifting problem for various classes of 4-cycle algebra direct systems. 1997 Academic Press Contents 1. Introduction. 2. Finite dimensional algebras. 3. Regular direct systems. 4. K 0 invariants. 5. Homology invariants I. 6. Lifting K 0 H 1 isomorphisms. 7. Unital even systems. 8. Unital odd symmetric systems. 9. Counterexample. 10. Homology invariants II. 11. The joint scale.