More on $(4,4)$ supermultiplets in $SU(2)\times SU(2)$ harmonic superspace

We define the $SU(2)\times SU(2)$ harmonic superspace analogs of tensor and nonlinear $(4,4)$, $2D$ supermultiplets. They are described by constrained analytic superfields and provide an off-shell formulation of a class of torsionful $(4,4)$ supersymmetric sigma models with abelian translational isometries on the bosonic target. We examine their relation to $(4,4)$ twisted multiplets and discuss different types of $(4,4)$ dualities associated with them. One of these dualities implies the standard abelian $T$-duality relations between the bosonic targets in the initial and dual sigma model actions. We show that $N=4$, $2D$ superconformal group admits a simple realization on the superfields introduced, and present a new superfield form of the $(4,4)$ $SU(2)\times U(1)$ WZNW action.