Lattice materials formed by hinged springs or elastic bonds may exhibit diverse anisotropy and asymmetry features of the overall elastic behavior depending on their unit cell configuration. The recently developed singum model transfers the force-displacement relationship of the springs in the lattice to the stress-strain relationship in the continuum particle, and provides the analytical form of tangential elasticity. When a pre-stress exists in the lattice, the stiffness tensor significantly changes due to the effect of the configurational stress. Different lattice structures lead to different symmetry of the stiffness tensors, which is demonstrated by five lattices. When all bonds exhibit the same length, regular hexagonal, honeycomb, and auxetic lattices demonstrate that the stiffness changes from an isotropic to anisotropic, from symmetric to asymmetric tensor. When the central symmetry of the unit cell is not satisfied, the primitive cell will contain more than one singums and the Cauchy-Born rule fails by the loss of equilibrium of the single singum. A secondary stress is induced to balance the singums, which may lead to the loss of minor symmetry of the stiffness tensor. Displacement gradient dij=uj,i is proposed to replace strain in the constitutive law for the general case because u1,2 and u2,1 may produce different stress states. Although the honeycomb lattice still exhibits isotropic behavior, for general auxetic lattices, an anisotropic and asymmetric elasticity is obtained with the loss of both minor and major symmetry, which is also demonstrated in a square lattice with unbalanced central symmetry and a chiral lattice. The modeling procedure and results may be generalized to three dimension and other lattices with the anisotropic and asymmetric stiffness.