Dual superconformal symmetry is a remarkable, hidden feature of N=4 SYM in 4 dimensions. Via AdS/CFT, such a symmetry corresponds to the invariance of the AdS(5) x S(5) superstring under specific combinations of bosonic and fermionic T-dualities. We show that AdS(d) x S(d) x S(d) superstrings with D(2,1;\alpha) isometry supergroup are T-self-dual if additional T-dualities along complexified S(d) directions are performed. This implies that CFTs dual to AdS(d) x S(d) x S(d) x T(10-3d) superstrings enjoy a new type of dual superconformal symmetry.