When you are talking about balanced headphones and headphone amps, it's a marketing gimmick. There is no such thing. The people that make this stuff use the term balanced, put xlr connectors on the gear and hope you don't ask too many questions. Here's what's really going on. Any time you see the work balanced, insert the word bridged. A balanced headphone amp is really a bridged amp. Bridging combines 2 channels into one, so your headphone amp has 4 channels. Single ended stereo signal transfer requires 3 conductors. Left and right channels and a common ground. You have 4 power amps and 3 wires. The incoming signal is converted to balanced inside the headphone amp. After the conversion, you now have 2 signals and 2 separate grounds and can make use of all 4 channels. The "balanced" output for your headphones usually requires a TRRS connector, or something else that supports 4 conductors. If your headphones have a hardwired connection, you can't use them on the output that's labeled balanced, and you can't bridge the amp. Most people think that the TRRS headphone jack labeled balanced, is balanced. Its not. That's why there is no such thing. Headphones are just a really small pair of speakers. If you want to call headphones balanced, you have to call the speakers in your living room balanced as well. You can only use your headphones if you have a detachable cable. Now that the ground is separated from the bridging process, each speaker in the headphone gets its own ground, just like any other pair of speakers. They get to use the term balanced because the signal had to be split at the input to separate the common ground. So, if you want your headphones to work properly, you can't bridge the amp.
The reason they try to hide this from you is there are some downsides to bridging amps. A bridged 2 channel amp sees half the resistance of a mono amp. For example, if you have an 8 ohm speaker, it becomes a 4 ohm load when you connect it to a bridged amp. You're forcing the amp to work twice as hard to drive the same load in unbridged mode. If you look at the specs on your headphone amp, you see the amp makes 2x the power in "balanced" mode. If your headphones are hard to drive, they may sound better with the amp bridged. If not, unbridged usually sounds better. If you want to use your amp in bridged mode, you'll need to get headphones that have a detachable cable, and a TRRS cable.