  The ability to test general relativity in extreme gravity regimes using
gravitational wave observations from current ground-based or future space-based
detectors motivates the mathematical study of the symmetries of black holes in
modified theories of gravity. In this paper we focus on spinning black hole
solutions in two quadratic gravity theories: dynamical Chern-Simons and scalar
Gauss-Bonnet gravity. We compute the principal null directions, Weyl scalars,
and complex null tetrad in the small-coupling, slow rotation approximation for
both theories, confirming that both spacetimes are Petrov type I. Additionally,
we solve the Killing equation through rank 6 in dynamical Chern-Simons gravity
and rank 2 in scalar Gauss-Bonnet gravity, showing that there is no nontrivial
Killing tensor through those ranks for each theory. We therefore conjecture
that the still-unknown, exact, quadratic-gravity, black-hole solutions do not
possess a fourth constant of motion.
