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We uncover field-theoretic underpinnings of symmetry relations formultifractal spectra at Anderson transitions and at critical points of otherdisordered systems. We show that such relations follow from the conformalinvariance of the critical theory, which implies their general character. Wealso demonstrate that for the Anderson localization problem the entireprobability distribution for the local density of states possesses a symmetryarising from the invariance of correlation functions of the underlyingnon-linear $\sigma$-model with respect to the Weyl group of the target space ofthe model.

Symmetries of multifractal spectra and field theories of Anderson localization