  We study fermionic T-duality symmetries of integrable Green-Schwarz
sigma-models on Anti-de-Sitter backgrounds with Ramond-Ramond fluxes,
constructed as Z_4 supercosets of superconformal algebras. We find three
algebraic conditions that guarantee self-duality of the backgrounds under
fermionic T-duality, we classify those that satisfy them and construct the map
of the monodromy matrix. We introduce new T-duality directions, where some of
them contain no bosonic directions, along which the backgrounds are self-dual.
We find that the only self-dual backgrounds are AdS_n x S^n for n=2,3,5. In
addition we find that the backgrounds AdS_n x S^1 for n=2,3,5, AdS_4 x S^2 and
AdS_2 x S^4 are self-dual at the level of the classical action, but have a
non-trivial transformation of the dilaton.
