  It is well known that the Kerr--NUT--AdS spacetimes possess hidden symmetries
encoded in the so-called principal Killing--Yano tensor. In this paper,
focusing on the four-dimensional case, we obtain a number of symmetry operators
for scalar, vector, and tensor perturbations, that are of degree two (to be
defined below) and homogeneous in the principal tensor. In particular, by
considering homogeneous operators that are linear, quadratic, and cubic in the
principal tensor, we recover a complete set of 4 mutually commuting operators
for scalar perturbations, underlying the separability of (massive) scalar wave
equation. Proceeding to vector and tensor perturbations of the Kerr--NUT--AdS
spacetimes, we find a set of 7 and 8 commuting operators, respectively. It
remains to be seen whether such operators can be used to separate the
corresponding spin 1 and spin 2 test field equations in these spacetimes.
