  World-sheet and spacetime supersymmetries that are manifest in some string
backgrounds may not be so in their T-duals. Nevertheless, they always remain
symmetries of the underlying conformal field theory. In previous work the
mechanism by which T-duality destroys manifest supersymmetry and gives rise to
non-local realizations was found. We give the general conditions for a 2-dim
N=1 supersymmetric sigma-model to have non-local and hence non-manifest
extended supersymmetry. We then examine T-duality as a mechanism of restoring
manifest supersymmetry. This happens whenever appropriate combinations of
non-local parafermions of the underlying conformal field theory become local
due to non-trivial world-sheet effects. We present, in detail, an example
arising from the model SU(2)/U(1) X SL(2,R)/U(1) and obtain a new exact 4-dim
axionic instanton, that generalizes the SU(2) X U(1) semi-wormhole, and has
manifest spacetime as well as N=4 world-sheet supersymmetry. In addition,
general necessary conditions for abelian T-duality to preserve manifest N=4
world-sheet supersymmetry are derived and applied to WZW models based on
quaternionic groups. We also prove some theorems for sigma-models with
non-local N=4 world-sheet supersymmetry.
