  We present covariant symmetry operators for the conformal wave equation in
the (off-shell) Kerr--NUT--AdS spacetimes. These operators, that are
constructed from the principal Killing--Yano tensor, its `symmetry
descendants', and the curvature tensor, guarantee separability of the conformal
wave equation in these spacetimes. We next discuss how these operators give
rise to a full set of mutually commuting operators for the conformally rescaled
spacetimes and underlie the $R$-separability of the conformal wave equation
therein. Then, by employing the WKB approximation to the `$\alpha$-modified
conformal wave equation' we derive an associated Hamilton--Jacobi equation with
a scalar curvature potential term and show its separability in the
Kerr--NUT--AdS spacetimes.
