  We present a novel derivation of the Teukolsky-Starobinsky identities, based
on properties of the confluent Heun functions. These functions define
analytically all exact solutions to the Teukolsky master equation, as well as
to the Regge-Wheeler and Zerilli ones. The class of solutions, subject to
Teukolsky-Starobinsky type of identities is studied. Our generalization of the
Teukolsky-Starobinsky identities is valid for the already studied linear
perturbations to the Kerr and Schwarzschild metrics, as well as for large new
classes of of such perturbations which are explicitly described in the present
article. Symmetry of parameters of confluent Heun's functions is shown to stay
behind the behavior of the known solutions under the change of the sign of
their spin weights. A new efficient recurrent method for calculation of
Starobinsky's constant is described.
