I’ve confronted several manufacturers on this, and every one of them reluctantly admitted I was right
I am 100% with @cleeds on this one, unless you can get Musical Fidelity to agree with you
.in writing.
Here is the confusion. The headphone amplifier outputs a fully balanced signal for headphones, using two 3-pin XLR connectors, one for each channel. Each channel uses two wires to carry the balanced signal, for a total of four active wires, probably in two separate cables.
Unfortunately, as a bit of a gimmick, it also allows a standard headphone jack to be inserted into the middle of either of the XLR connectors. I presume this is a three-section jack designed to carry a non-balanced stereo signal over three wires (left, right and common). So when the OP plugs his jack in, he gets unbalanced stereo. My educated guess is that it does not matter which side he uses (left channel may become right, who knows)!
To get the undoubted benefits of truly balanced operation, he needs to rewire each side of his headphones directly to its own 3-pin XLR connector and completely remove any jack.
It is London to a brick that when Musical Fidelity says one of its products is fully balanced, its active circuitry is doubled from input to output.
Ideally the attached player should output in balanced mode too. Then the signal path from the output of the player to the coils of the headphone will be fully balanced.
Note: balanced means the signal is duplicated on two paths, where one path is the opposite polarity of the other. The combined signal is twice the magnitude of an unbalanced signal. More importantly, Electro-Magnetic Interference (EMI) acts equally and oppositely on each path, and is cancelled out in the combined signal.

