The Woo Audio WA22 is a fully balanced amp with both balanced inputs and balanced outputs. The confusion regarding the WA22 stems from the confusion people have between the terms "balanced" and "differential".
For the purposes of this discussion:
A balanced circuit only means that you have two signal conductors, each with the same impedance to circuit ground.
A balanced, differential circuit means you have two balanced conductors, each with the same signal except on one conductor the signal is inverted (180 degrees out of phase).
Each channel of the WA22 has two signal inputs, + (in-phase signal) and - (inverted-phase signal) , (balanced and requiring a differential signal pair) and a ground. Each of the signal inputs feeds a separate drive circuit to either side of the transformer primary.
This creates a balanced, differential drive circuit that works by applying an amplified version of the original audio signal (+) to one side of the transformer primary while simultaneously applying an amplified inverted version of the original audio signal (-) to the other side of the transformer primary. Driving both sides of the transformer primary with a differential circuit doubles the slew rate (rise time) of the amplifier, and reduces noise induced from the external balanced cable run through the internal circuit. Actively controlling both sides of the transformer primary should also reduce distortion (grounds do float) but some may argue that it is negated by having dual drive circuits - each contributing distortion components.
While the WA22 will work fine with a single-ended input (rca), half of the drive circuit stays dormant at 0 Volts (or floats around a little) and the true audio performance of the WA22 remains underutilized.
The outputs of the WA22 transformers are balanced - all conductors (output signals and ground) have the same impedance to circuit ground. When I bought my WA22, I asked Jack Woo about the issue and he advised that while the XLR and 1/4" outputs share a common ground, the output ground is not connected to circuit ground - hence the output conductors are balanced.
There is definitely an advantage (audio performance) to using a balanced differential circuit to drive the transformer primary but it is unclear what advantages one would gain with a differential output from the transformer.
What comes out of the WA22 transformers is one left-channel signal, one right-channel signal, and a common ground. You can connect to the output via 1/4" TRS, dual 3-pin XLR, or single 4-pin XLR. It is unclear what advantage a balanced HP cable would make as the XLR outputs are electrically the same as the 1/4" TRS.
This is no slight against the WA22. It is a very elegant design. As I mentioned, it is unclear what audio advantage would be gained with a differential output. The WA22 is an ultra high-performance audio amplifier with balanced differential inputs, class A differential amplifier circuits, and balanced single-ended outputs.