The identification of multiple interacting inclusions with uniform internal stresses in an infinite elastic matrix subjected to a uniform remote loading is of fundamental importance in the mechanics and design of particulate composite materials. In anti-plane shear and plane deformations, certain sufficient conditions have been established in the literature which guarantee uniform internal stresses inside multiple interacting inclusions displaying various symmetries when the matrix is subjected to specific uniform remote loading. Correspondingly, sufficient conditions which allow for the design of multiple interacting inclusions independent of any specific form of (uniform) remote loading have also been established. In this paper, we demonstrate rigorously that, in all cases, these sufficient conditions are also necessary conditions and indeed allow for the identification of all possible collections of such inclusions.