  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.
