We formulate a new version of the local Langlands correspondence
for discrete L-parameters which involves (Weyl orbits of) packets of
representations of all twisted Levi subgroups of a connected reductive
group G through which a given parameter factors and prove that this version
of the correspondence is true if one assumes the pre-existing local
Langlands conjectures. Twisted Levi subgroups are crucial objects in the
study of supercuspidal representations; this work is a step towards
deepening the relationship between the representation theory of p-adic
groups and the Langlands correspondence. This talk will also serve as a
rough introduction to the refined local Langlands correspondence (and its
generalizations). This is joint work with David Schwein.