Motivated by the black hole stability problem, we discuss the structure of linear test fields on Kerr spacetime. The dynamics of the linearized gravitational field on a Kerr background is governed by certain curvature components solving the Teukolsky master equations (TME) and Teukolsky-Starobinski identities (TSI). I will show that the TME and TSI operators are self-adjoint in an appropriate sense. Combined with a result due to Wald from 1978, this leads to symmetry operators of the linearized Einstein equations of order four and six, respectively. Interpretations and possible applications will be discussed. The results were derived using spinors and advanced symbolic computer algebra tools for xAct, which I will introduce briefly.