  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.
