Suppose G is a real reductive algebraic group, is an automorphism of G, and is a quasicharacter of the group of real points G(R). Under some additional assumptions, the theory of twisted endoscopy associates to this triple real reductive groups H. The Local Langlands Correspondence partitions the admissible representations of H(R) and G(R) into L-packets. The author proves twisted character identities between L-packets of H(R) and G(R) comprised of essential discrete series or limits of discrete series.