We study the properties of conformal operators in the SL(2) sector of planar N = 4 SYM and its supersymmetric SL(2 vertical bar 2) extension. The correlation functions of these operators and their form factors with respect to asymptotic on-shell states are determined by two different polynomials which can be identified as eigenstates of the dilatation operator in the coordinate and momentum representations, respectively. We argue that, in virtue of integrability of the dilatation operator, the two polynomials satisfy a duality relation - they are proportional to each other upon an appropriate identification of momenta and coordinates. Combined with the conventional N = 4 superconformal symmetry, this leads to the dual superconformal symmetry of the dilatation operator. We demonstrate that this symmetry is powerful enough to fix the eigenspectrum of the dilatation operator to the lowest order in the coupling. We use the relation between the one-loop dilatation operator and Heisenberg spin chain to show that, to lowest order in the coupling, the dual symmetry is generated by the Baxter Q-operator in the limit of large spectral parameter. (C) 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).