The Teukolsky equation possesses discrete symmetries that constrain the properties of black hole perturbations and their quasinormal mode spectrum. In this study, we explore how a class of modifications of the Teukolsky potential can alter the symmetry structure of the equation and break the $m=0$ degeneracy of quasinormal modes. We prove this result in the frequency domain using the master Teukolsky equation and in the time domain via ($2+1$)-dimensional simulations. We also show that the discrete symmetries can be leveraged for a more efficient characterization of the $m=0$ quasinormal modes from time-domain evolutions. As a theory-specific application, we consider the case of higher-derivative theories of gravity, highlighting that time-domain implementations of frequency-domain potentials can give rise to additional nonphysical branches of modes.

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• Alternative gravity theories
• Classical black holes
• General relativity
• Gravitational waves