  Irreversibility, crucial in both thermodynamics and information theory, is
naturally studied by comparing the evolution -- the (forward) channel -- with
an associated reverse -- the reverse channel. There are two natural ways to
define this reverse channel. Using logical inference, the reverse channel is
the Bayesian retrodiction (the Petz recovery map in the quantum formalism) of
the original one. Alternatively, we know from physics that every irreversible
process can be modeled as an open system: one can then define the corresponding
closed system by adding a bath ("dilation"), trivially reverse the global
reversible process, and finally remove the bath again. We prove that the two
recipes are strictly identical, both in the classical and in the quantum
formalism, once one accounts for correlations formed between system and the
bath. Having established this, we define and study special classes of maps:
product-preserving maps (including generalized thermal maps), for which no such
system-bath correlations are formed for some states; and tabletop
time-reversible maps, when the reverse channel can be implemented with the same
devices as the original one. We establish several general results connecting
these classes, and a very detailed characterization when both the system and
the bath are one qubit. In particular, we show that when reverse channels are
well-defined, product-preservation is a sufficient but not necessary condition
for tabletop reversibility; and that the preservation of local energy spectra
is a necessary and sufficient condition to generalized thermal operations.
