  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 conformally invariant mutually commuting operators for the conformally
rescaled spacetimes and underlie the $R$-separability of the conformal wave
equation therein. Finally, by employing the WKB approximation we derive the
associated Hamilton-Jacobi equation with a scalar curvature potential term and
show its separability in the Kerr-NUT-AdS spacetimes.
