A modular category C gives rise to a differential graded modular functor which assigns to the torus the Hochschild complex and, in the dual description, the Hochschild cochain complex of C. On both complexes, the monoidal product of C induces the structure of an E2-algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in C a Calabi-Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of C with a second E2-structure. Our main result is that the action of a specific element S in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second E2-structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of C. In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element S. In the semisimple case, both results reduce to the Verlinde formula. In the non-semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non-semisimple generalizations of the Verlinde formula.