In this paper, we identify the Ad-equivariant twisted K-theory of a compact Lie group G with the “Verlinde group” of isomorphism classes of admissible representations of its loop groups. Our identification preserves natural module structures over the representation ring R(G) and a natural duality pairing. Two earlier papers in the series covered foundations of twisted equivariant K-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free π1. Here, we recall the earlier material as needed to make the paper self-contained. Going further, we discuss the relation to semi-infinite cohomology, the fusion product of conformal field theory, the rôle of energy and a topological Peter-Weyl theorem.