  Recent research suggests that when a system has a "false time reversal
violation" the Onsager reciprocity relations hold despite the presence of a
magnetic field. The purpose of this work is to clarify that the Onsager
relations may well be violated in presence of a "false time reversal
violation": that rather guarantees the validity of distinct relations, which we
dub "false Onsager relations". We also point out that for quantum systems
"false time reversal violation" is omnipresent and comment that, per se, this
has in general no consequence in regard to the validity of Onsager relations,
or the more general non-equilibrium fluctuation relations, in presence of a
magnetic field. Our arguments are illustrated with the Heisenberg model of a
magnet in an external magnetic field.
