Double Symmetries in Field Theories

In the paper a concept of a double symmetry is introduced, and its qualitative characteristics and rigorous definitions are given. We describe two ways to construct the double-symmetric field theories and present an example demonstrating the high efficiency of one of them. In noting the existing double-symmetric theories we draw attention at a dual status of the group $SU(2{)}_L\otimes SU(2{)}_R$ as a secondary symmetry group, and in this connexion we briefly discuss logically possible aspects of the $P$-violation in weak interactions.