  Recent studies of scattering amplitudes in planar N=4 SYM theory revealed the
existence of a hidden dual superconformal symmetry. Together with the
conventional superconformal symmetry it gives rise to powerful restrictions on
the planar scattering amplitudes to all loops. We study the general form of the
invariants of both symmetries. We first construct an integral representation
for the most general dual superconformal invariants and show that it allows a
considerable freedom in the choice of the integration measure. We then perform
a half-Fourier transform to twistor space, where conventional conformal
symmetry is realized locally, derive the resulting conformal Ward identity for
the integration measure and show that it admits a unique solution. Thus, the
combination of dual and conventional superconformal symmetries, together with
invariance under helicity rescalings, completely fixes the form of the
invariants. The expressions obtained generalize the known tree and one-loop
superconformal invariants and coincide with the recently proposed coefficients
of the leading singularities of the scattering amplitudes as contour integrals
over Grassmannians.
