  A set of classical or quantum states is equivalent to another one if there
exists a pair of classical or quantum channels mapping either set to the other
one. For dichotomies (pairs of states), this is closely connected to (classical
or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a
RD remains unchanged when a channel is applied to the dichotomy, then there is
a recovery channel mapping the image back to the initial dichotomy. Here, we
prove for classical dichotomies that equality of the RDs alone is already
sufficient for the existence of a channel in any of the two directions and
discuss some applications. In the quantum case, all families of quantum RDs are
seen to be insufficient because they cannot detect anti-unitary
transformations. Thus, including anti-unitaries, we pose the problem of finding
a sufficient family. It is shown that the Petz and maximal quantum RD are still
insufficient in this more general sense and we provide evidence for sufficiency
of the minimal quantum RD. As a side result of our techniques, we obtain an
infinite list of inequalities fulfilled by the classical, the Petz quantum, and
the maximal quantum RDs. These inequalities are not true for the minimal
quantum RDs. Our results further imply that any sufficient set of conditions
for state transitions in the resource theory of athermality must be able to
detect time-reversal.
