N=(4,4), 2D supergravity in SU(2)xSU(2) harmonic superspace

We work out the basics of conformal $N=(4,4)$, 2D supergravity in the $N=(4,4)$, 2D analytic harmonic superspace with two independent sets of harmonic variables. We define the relevant most general analytic superspace diffeomorphism group and show that in the flat limit it goes over into the ``large'' $N=(4,4)$, 2D superconformal group. The basic objects of the supergravity considered are analytic vielbeins covariantizing two analyticity-preserving harmonic derivatives. For self-consistency they should be constrained in a certain way. We solve the constraints and show that the remaining irreducible field content in a WZ gauge amounts to a new short $N=(4,4)$ Weyl supermultiplet. As in the previously known cases, it involves no auxiliary fields and the number of remaining components in it coincides with the number of residual gauge invariances. We discuss various truncations of this ``master'' conformal supergravity group and its compensations via couplings to $N=(4,4)$ superconformal matter multiplets. Besides recovering the standard minimal off-shell $N=(4,4)$ conformal and Poincar\'e supergravity multiplets, we find, at the linearized level, several new off-shell gauge representations.