  We provide a systematic and comprehensive derivation of the linearized
dynamics of massive and partially massless spin-2 particles in a Schwarzschild
(anti) de Sitter black hole background, in four and higher spacetime
dimensions. In particular, we show how to obtain the quadratic actions for the
propagating modes and recast the resulting equations of motion in a
Schr\"odinger-like form. In the case of partially massless fields in
Schwarzschild de Sitter spacetime, we study the isospectrality between modes of
different parity. In particular, we prove isospectrality analytically for modes
with multipole number $L=1$ in four spacetime dimensions, providing the
explicit form of the underlying symmetry. We show that isospectrality between
partially massless modes of different parity is broken in higher-dimensional
Schwarzschild de Sitter spacetimes.
