  We uncover field-theoretic underpinnings of symmetry relations for
multifractal spectra at Anderson transitions and at critical points of other
disordered systems. We show that such relations follow from the conformal
invariance of the critical theory, which implies their general character. We
also demonstrate that for the Anderson localization problem the entire
probability distribution for the local density of states possesses a symmetry
arising from the invariance of correlation functions of the underlying
non-linear $\sigma$-model with respect to the Weyl group of the target space of
the model.
