We constructed gamma operators which are superalgebraic analogs of the Dirac gamma matrices as well as two additional gamma operators which have no analogs in the Dirac theory. We found a new mechanism of the left-right symmetry breaking which is absent in the usual theory of Dirac spinors. We constructed Lorentz invariant gamma operators from operators of creation and annihilation of spinors. These gamma operators are also analogs of the Dirac gamma matrices, however they are not related to Lorentz transformations but generate vector fields as affine spinor connections. We have shown that the theory is equivalent to an extended version of the Pati-Salam theory.