We investigate equivariant analogues of the Minkowski--Weyl theorem and Gordan's lemma in an infinite-dimensional setting, where cones and monoids are invariant under the action of the infinite symmetric group. Building upon the framework developed earlier, we extend the theory beyond the nonnegative case. Our main contributions include a local equivariant Minkowski--Weyl theorem, local-global principles for equivariant finite generation and stabilization of symmetric cones, and a full proof of the equivariant Gordan's lemma. We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.