The boundary element methods are suited for the analysis of symmetric and antisymmetric problems - in which only a part (half, quadrant or octant) of the structure needs to be explicitly considered - since, as an additional advantage when compared with a domain discretization method, no interpolation is required along the symmetry axes (for 2D problems) or planes (for 3D problems) and, consequently, no approximations are introduced thereon. Although such computational simplification may prevent some of the structures allowable rigid body movements (elasticity problems considered), this fact may be completely ignored as concerning the implementation of the traditional (collocation or Galerkin) boundary element methods. In the hybrid boundary element methods, on the other hand, special orthogonality conditions, directly or indirectly related to rigid body displacements, are required for the evaluation of elements about the main diagonal of some matrices (flexibility, displacement and stress matrices). Then, a central issue in such methods is the assessment of these matrices spectral properties for any combination of symmetry and antisymmetry and, most important, the investigation of conceptually equivalent, substitutive properties. As presented in this work, the hybrid boundary element methods, although based on singular Green s functions, are able to simulate, in terms of both virtual work and field interpolation, the simplest stress states. Then, one demonstrates that for every missing rigid body displacement - brought about by some symmetry or antisymmetry consideration - one may lay hold of a simple (in most cases constant) stress state, which enables establishing appropriate spectral properties. This work introduces the underlying variational concepts of the hybrid boundary element method and outlines the special consideration of simple (polynomial) stress states, as generally formulated for 3D elasticity, since 2D elasticity and problems of potential may be dealt with as particular cases. All combinations of symmetry and antisymmetry are outlined with the aim of numerical implementation. A series of 2D examples for problems of potential illustrate the theoretical