Diversity of off--Shell Twisted (4,4) Multiplets in SU(2) x SU(2) Harmonic Superspace

We elaborate on four different types of twisted ${\cal N}=(4,4)$ supermultiplets in the $SU(2) \times SU(2)$, 2D harmonic superspace. In the conventional ${\cal N}=(4,4)$, 2D superspace they are described by the superfields $\hat q^{i a}$, $\hat q^{\und i a}$, $\hat q^{i \und a}$, $\hat q^{\und i \und a}$ subjected to proper differential constraints, $(i, \und i, a, \und a)$ being the doublet indices of four groups SU(2) which form the full R-symmetry group $SO(4)_L\times SO(4)_R$ of ${\cal N}=(4,4)$ supersymmetry. We construct the torsionful off--shell sigma model actions for each type of these multiplets, as well as the corresponding invariant mass terms, in an analytic subspace of the $SU(2) \times SU(2)$ harmonic superspace. As an instructive example, ${\cal N}=(4,4)$ superconformal extension of the $SU(2) \times U(1)$ WZNW sigma model action and its massive deformation are presented for the multiplet $\hat q^{i \und a}$ . We prove that ${\cal N}=(4,4)$ supersymmetry requires the general sigma model action of pair of different multiplets to split into a sum of sigma model actions of each multiplet. This phenomenon also persists if a larger number of non-equivalent multiplets are simultaneously included. We show that different multiplets may interact with each other only through mixed mass terms which can be set up for multiplets belonging to ``self-dual'' pairs $(\hat q^{i a}, \hat q^{\und i \und a})$ and $(\hat q^{\und i a}, \hat q^{i \und a})$ . The multiplets from different pairs cannot interact at all. For a ``self-dual'' pair of the twisted multiplets we give the most general form of the on-shell scalar potential.