Irreducible symmetry operators (ISO’s) are defined and their properties are displayed. A method for the systematic construction of ISO’s in the form of products of interchangeable factors is described. The key to the method is a set of conditions for directly testing for the irreducibility of induced symmetry operators formulated in terms of the operators themselves. Detailed examples are presented for the tetrahedral group and double group. The tetrahedral double group provides an example in which the induction process in its simplest form fails, but which does yield to a modification of the induction techniques.