  We consider the distribution $P(\phi)$ of the Hatano-Sasa entropy, $\phi$, in
reversible and irreversible processes, finding that the Crooks relation for the
ratio of the pdf's of the forward and backward processes,
$P_F(\phi)/P_R(-\phi)=e^{\phi}$, is satisfied not only for reversible, but also
for irreversible processes, in general, in the adiabatic limit of "slow
processes." Focusing on systems with a finite set of discrete states (and no
absorbing states), we observe that two-state systems always fulfill detailed
balance, and obey Crooks relation. We also identify a wide class of systems,
with more than two states, that can be "coarse-grained" into two-state systems
and obey Crooks relation despite their irreversibility and violation of
detailed balance. We verify these results in selected cases numerically.
