  We uncover hidden spectral symmetries of the Teukolsky equation in Kerr(-de
Sitter) black holes, recently conjectured by Aminov, Grassi and Hatsuda (arXiv:
2006.06111 and arXiv: 2007.07906, 2020) in the zero cosmological constant case.
Using these symmetries, we provide a new, simpler proof of mode stability for
subextremal Kerr black holes. We also present a partial mode stability result
for Kerr-de Sitter black holes.
