  In this paper we study skew Poincar\'e-Birkhoff-Witt extensions over weak
symmetric and $(\Sigma,\Delta)$-weak symmetry rings. Since these extensions
generalize Ore extensions of injective type and another noncommutative rings of
polynomial type, we unify and extend several results in the literature
concerning the property of being symmetry. Under adequate conditions, we
transfer the property of being weak symmetric or $(\Sigma,\Delta)$-weak
symmetric from a ring of coefficients to a skew PBW extension over this ring.
We illustrate our results with remarkable examples of algebras appearing in
noncommutative algebraic geometry and theoretical physics.
