  A new $H(\textrm{divdiv})$-conforming finite element is presented, which
avoids the need for super-smoothness by redistributing the degrees of freedom
to edges and faces. This leads to a hybridizable mixed method with
superconvergence for the biharmonic equation. Moreover, new finite element
divdiv complexes are established. Finally, new weak Galerkin and $C^0$
discontinuous Galerkin methods for the biharmonic equation are derived.
