I can only offer my own thoughts based on some thermodynasmics, but the following seems very reasonable to me:
The class of pertubation you allready know is the adiabatic one. IF we assume reversibility, adiabatic processes are exactly isentropic (fixed entropy) ones. It makes therefore sense to divide the perturbations into two classes:
other (non isentropic) where entropy changes.
Furthermore, as δQ = TdS (for the reversible heat transfere δQ), every heat tranfere has to involve either temperature or entropy change. It makes therefore sense to decompose every non vanishing perturbation into:
ADIABATIC/ISENTROPIC part, necessarily non isothermal
ISOTHERMAL part, necessarily non-adiabatic
And in fact I think these are two complementary classes studied in cosmology.
I hope this helped,