T cells recognize with a single receptor both a product of antigens processed by antigen presenting cells (APC1) and a self-marker molecule, encoded by the major histocompatibility complex (MHC, a property termed MHC-restricted recognition of antigen). During their differentiation in the thymus, T cells "learn" what to regard as self-MHC molecules, and only the cells once able to recognize antigen in the context of self-MHC will be "positively selected" to exit the thymus. The cells, once capable of reacting to self molecules, do not exit the thymus. They are "negatively selected" (deleted). Both "positive" and "negative" selection depends on the T-cell-receptor (TCR) specificity. Furthermore, the TCR specificity determines the final phenotype of the mature T cells; namely, the cells with receptors specific for the MHC-class I molecule will acquire the CD4-CD8+ phenotype, while the cells with receptors specific for the MHC-class II molecule will acquire the CD4+CD8- phenotype. However, a few mature T cells in the periphery do not follow the rule: CD4 expression class II restriction and CD8 expression class I restriction. We believe that these T lymphocytes have a receptor with very high affinity for one class of MHC molecules and cross-react with another class of MHC molecules (with somewhat lower affinity). The majority of T lymphocytes with such receptors bind the thymic MHC molecule, for which they have the highest affinity. Since this affinity is too high for further differentiation, such clones are deleted in the thymus. However, a small fraction of these cells bind the alternative class of MHC molecules, due to cross-reactivity of their receptors.(ABSTRACT TRUNCATED AT 250 WORDS)