We introduce and study a new notion of amenability called symmetric pseudo-amenability. We obtain some properties of symmetrically pseudo-amenable Banach algebras and with examples, we compare this type of amenability with some other types of amenability. We also provide some special classes of symmetrically pseudo-amenable Banach algebras. Finally, Jordan derivations and Lie derivations from a class of Banach algebras into appropriate Banach bimodules are investigated using the notion of symmetric pseudo-amenability.
