When talking about "balanced" headphones and any kind of similar headphone amp, we are essentially referring to a differential amplifier or differential signaling. If you think about an old two-channel car stereo, you could bridge each channel together and get double the output/power. It might not always be double the power output (it really depends on the circuit), but ultimately, you've increased the overall output. You've also made the circuit more efficient. Not only is it more powerful but it's also a lot quieter because it operates at a lower load capacity on the power supply, etc. Differential amplifiers are a type of operational amplifiers, which are a core component of achieving this balanced circuitry.
Any headphone amp is what we call a differential amplifier circuit. It's not common-mode rejection. You can't do common-mode rejection with a headphone because it's a speaker. There's no send and receive - you're just sending the signal to the headphone - so it's different. People get very confused about that.
Common-mode rejection can also work as a splitter to make the lines quieter. Not a lot of products are "truly" balanced. You'll see more headphone amps that are truly balanced versus speaker amps, cd players, or preamps because it's more expensive to create truly balanced products.
If we’re being completely honest here, most headphone amplifiers are designed with pseudo-balanced circuits as a cost-saving method. To make a “true” balanced circuit, you would have to double all the circuitry from input to output. By doubling the circuitry, you are also doubling the cost to produce. If the one clear answer we have on this page is that true balanced circuit design will always sound better than single-ended, and if the cost wasn’t an issue, then everyone would design their amps the same way. But the cost is always a factor because when something is expensive to make, it’s also expensive to buy! Again, this only applies to true balanced circuit design, which is rare in the market.
An amplifier provides output power to a set of headphones. Depending on your application or set up, it might be possible that you need additional power, say if you have headphones that have power-hungry drivers. For example, if you are listening to music on your phone but don’t have enough juice to power your headphones, it will result in very poor audio quality.
It is also important to keep in mind what kind of connections your headphones have. If you have a balanced connection (XLR) on your headphones but a quarter-inch output on your headphone amplifier, it won't work, unless you have a compatible cable. Here at Moon Audio, we can create custom solutions for your headphones. Again, it is not your headphones that are balanced or unbalanced, but the connection to the speakers. These cables and connectors can be switched out with various options depending on if the stock option is not your preference. Different connectors, impedance ratings, and other factors that will be crucial to factor in when deciding what kind of headphone amplifier you will need.
Here are some examples of single-ended and balanced headphone amp options: