I keep seeing a clearly greater 2nd harmonic product from unbalanced input than from balanced input. And somewhat higher 3rd harmonic from balanced input than from unbalanced input.
This is expected.
Even-order distortion (H2, H4, ...) is asymmetrical. With H2, a sinewave might get rounded off at the top but sharper at the bottom.
For a moment, let's a treat a balanced input like a pair of unbalanced ones, one of which is subtracted from the other. (Side note: That actually works decently if you need a mono balanced input in a pinch but unbalanced stereo is all you have. It often has to be done manually, but e.g. REW supports this for both inputs and outputs.)
Let's say they both distort the signal in such a way that +1 comes out as +0.9 and -1 comes out as -1.1.
Now we'll feed this input with a BTL signal: +1 becomes +1 on the hot leg and -1 on the cold leg. In an ideal distortion-free input, we would expect a differential of +1 -(-1) = +2.
+ input (hot): +1 in --> +0.9 out
- input (cold): -1 in --> -1.1 out
-----------------------------------
Differential: +0.9 - (-1.1) = +2.
The distortion has effectively been cancelled!
The same does not apply to
odd-order distortion like H3, which you might see as symmetrical rounding of the sinewave tops. When both +1 and -1 become +0.9 and -0.9 equally, there is nothing the input can do about this. So you can only hope for H3 to
not increase at best. It may go up the balanced signal is at higher absolute levels, including after the balanced receiver.
Note that while I have lumped together all the distortion in the input in this example, in general it's actually the product of the entire analog signal chain.