In case it might be helpful, here's some information I had collected for myself previously in my research on the topic. I'm including some choice quotes, but I recommend going to the links to get the full story on each.
On Dec 5, 2015 Tyll Herstens gets into the nitty gritty of whether a balanced headphone amp should even be called balanced and takes some responsibility for the situation:
http://www.superbestaudiofriends.or...iscussion-for-noobs-and-boobs.733/#post-18938
"To some extent, I'm responsible for this word showing up in the world of headphones with the introduction of the Blockhead Many years ago. Bottom line: I don't think we can ever use this word in a technically correct way as it applies to balanced headphone amps. I guess I have to side with Steve E.'s take on the subject...technically...that 'bridged' is the more correct term---but that horse left the barn long ago..."
On Dec 6, 2017 Bob Katz had this to say:
https://www.innerfidelity.com/content/katzs-corner-episode-21-how-insensitive-part-2
"You've probably heard this before, but it bears repeating:
There is no such thing as a balanced headphone, because a headphone is not an active device, it has no innate reference to ground. A better term for this type circuit would be "symmetrical push-pull". I am agnostic about balanced versus unbalanced headphone amps. If an amplifier can deliver the sonics at the required levels, then it should not matter if it's balanced or unbalanced. A balanced amp can produce 6 dB more level with the same power supply as the unbalanced. But the unbalanced amp's designer can use a larger power supply and achieve equal level. Sure, it's nice to say that an amp like the Mjolnir is fully symmetrical from input to output, but does that translate to a sonic advantage or just an engineering goal? I would never dare to claim an amplifier's sonic superiority is due to its balanced topology. The variables cannot be isolated..."
His Episode 22 which will have the measurements from his amp roundup are still pending (he's skipped to 'Episode 23' for now).
The CEO of Lake People / Vioelectric put together this paper with some schematics, where he says:
http://violectric-usa.com/download/Balanced Headphone Amplifier Explained.pdf
"'Balanced' amps are also called Push-Pull Amps or BTL amps (Bridge-Terminated-Load)."
If you go looking, there are lots of websites out there were people document mods to headphones to get a 'balanced' cable to them, such as this:
https://robrobinette.com/HeadphoneMods.htm
Also worth noting is that only audiophiles throw around the term 'fully balanced amp', which isn't something an electronics engineer would recognize as having any meaning. Example in use:
http://www.6moons.com/audioreviews/balancedheadphones/one.html
Here's an example of the type of unsubstantiated here-say floating around on the topic as well:
http://www.instructables.com/id/Balanced-Headphones-for-About-130/
"Balanced mode delivers 4x the power to your voice coils, cuts your signal-to-noise ratio by 4, doubles the signal slew rate, and eliminates channel-to-channel crosstalk. The result is a precipitous drop in your noise floor, almost unbelievable increase in audio detail, much fuller and precisely defined soundstage and imaging, and an overall warmer, richer and fuller sound reaching your ears."
Hah! And for those into trivia, Moon Audio (which has as one line of their business headphone cables) lists quite a few of the balanced connector types:
https://www.moon-audio.com/single-ended-versus-balanced-connection
- Dual 3pin XLR connectors
- Single 4 pin XLR
- RSA/ALO Male Connector
- 2.5mm TRRS
- 3.5mm TRRS mini
- iBasso balanced
- JHAudio JH3a connector
However they missed the Japan Electronics and Information Technology Industries Association 4.4mm balanced connection standard that was introduced in 2014:
https://www.headfonia.com/picture-sunday-4-4mm-balanced-effect-audio-ares-ii/
Anyway, lots of cable options out there. Since my only balanced amp is in my Onkyo Rubato DAP which has a 2.5mm TRRS, that's what I've been purchasing with regards to cables. Obviously you can convert to XLR with something like this:
https://www.ebay.com/itm/4-pin-XLR-...FI-Cable-headphone-Audio-Adapter/202182913527
Aftermarket balanced cables for Hifiman HE-400i (and similar types of headphones that have 2.5mm TRS sockets on the cups) start at around $43:
https://www.ebay.com/itm/2-5mm-Trrs...400S-HE-400I-HE560-HE-350-HE1000/202122585731
The Fiio RC-MMCXB cable goes for about $31:
http://www.fiio.net/en/products/58
I made the mistake of thinking that the Fiio cable would work with my Onkyo ES-CTI300 headphones (which have MMCX connectors on the cups) but they had a loose fit. Cable worked fine on a couple of IEMs I have though. However for $15.50 I was able to get the "19 core 2.5mm balanced" MMCX cable here which was advertised as compatible with the ES-CTI300 and sure enough it works great:
https://www.ebay.com/itm/MMCX-Silve...Cable-3-5mm-Audio-2-5mm-Balanced/262966678773
I use a Fiio L26 adapter to use my headphones with balanced cables on an unbalanced source. The nice aspect about 2.5mm TRRS balanced becoming somewhat common and downmarket is that prices tend to be lower than for the XLR stuff. The Fiio F5 IEMs come with both an unbalanced and balanced MMCX cable (along with a full suite of accessories) for $60 street. The Pioneer SE-MHR5 over-ear headphones went on special for $80 during the 2017 holidays and come with both a balanced (2.5mm TRRS) and unbalanced cable.
I'm really looking forward to a review of balanced vs. unbalanced amps. If you're ever looking for some headphones to torture-test amps, the ones I know of offhand with the worst impedance variations are the Audio Technica ATH-R70x (peaks at over 900 Ohm):
https://www.innerfidelity.com/images/AudioTechnicaATHR70x.pdf
And in the 20-year vintage category, the Aurex HR-V9 gets over 4000 Ohm:
https://www.innerfidelity.com/images/AurexHRV9.pdf