  Context: Discrete symmetries have found numerous applications in photonics
and quantum mechanics, but remain little studied in fluid mechanics,
particularly in astrophysics. Aims: We aim to show how PT and anti-PT
symmetries determine the behaviour of linear perturbations in a wide class of
astrophysical problems. They set the location of Exceptional Points in the
parameter space and the associated transitions to instability, and are
associated to the conservation of quadratic quantities that can be determined
explicitly. Methods: We study several classical local problems: the
gravitational instability of isothermal spheres and thin discs, the
Schwarzschild instability, the Rayleigh-B\'enard instability and acoustic waves
in dust-gas mixtures. We calculate the locations and the order of the
Exceptional Points with a method of resultant, as well as the conserved
quantities in the different regions of the parameter space using Krein theory.
Results: All problems studied here exhibit discrete symmetries, even though
Hermiticity is broken by different physical processes (self-gravity, buoyancy,
diffusion, drag). This analysis provides genuine explanations for certain
instabilities, and for the existence of regions in the parameter space where
waves do not propagate. Those correspond to breaking of PT and anti-PT
symmetries respectively. Not all instabilities are associated to symmetry
breaking (e.g. the Rayleigh-Benard instability).
