  A sigma model with four-dimensional target space parametrized by chiral and
twisted chiral N=(2,2) superfields can be extended to N=(4,4) supersymmetry
off-shell, but this is not true for a model of semichiral fields, where the
N=(4,4) supersymmetry can only be realized on-shell. The two models can be
related to each other by T-duality.
  In this paper we perform a duality transformation from a chiral and twisted
chiral model with off-shell N=(4,4) supersymmetry to a semichiral model. We
find that additional non-linear terms must be added to the original
transformations to obtain a semichiral model with N=(4,4) supersymmetry, and
that the algebra closes on-shell as a direct consequence of the T-duality.
