  We perform a thorough study of greybody factors for minimally-coupled scalar
fields propagating on the background of rotating black holes in higher (odd)
dimensions with all angular momenta set equal. For this special case, the
solution enjoys an enhanced symmetry, which translates into the advantageous
feature of being cohomogeneity-1, i.e., these backgrounds depend on a single
radial coordinate. Our analysis contemplates three distinct situations, with
the cosmological constant being zero, positive or negative. Using the technique
of matched asymptotic expansions we compute analytically the greybody factors
in the low-frequency regime, restricting to s-wave scattering. Our formulas
generalize those obtained previously in the literature for the static and
spherically symmetric case, with corrections arising from the change in the
horizon area due to rotation. It is also proven that, for this family of black
holes, the horizon area is a decreasing function of the spin parameter, without
regard of dimensionality and of cosmological constant. Through an improvement
on a calculation specific to the class of small black holes in anti-de Sitter
and not restricted to the usual low-frequency regime, we uncover a rich
structure of the greybody spectrum, more complex than previously reported but
also enjoying a certain degree of universality. We complement our low-frequency
analytic results with numerical computations valid over a wide range of
frequencies and extend them to higher angular momentum quantum numbers,
$\ell>0$. This allows us to probe the superradiant regime that is observed for
corotating wavefunctions. We point out that the maximum amplification factor
for intermediate-size black holes in anti-de Sitter can be surprisingly large.
