Symmetrical characteristics of elastic structures will change when different tensioncompression moduli was employed. Based on the necessary conditions satisfied by stress and displacement, we analyzed symmetry and antisymmetry under the principal stress state using a computational example. The research indicates that antisymmetry of elastic structures in the magnitude and direction of principal stress will change due to the nonlinearity of materials while symmetry still holds, and the superposition theorem will fail when a complex problem is resolved into symmetric and antisymmetric components. It is concluded that the principal stress state due to the bimodular constitutive model based on principal direction is key.