It has to do with the phase of the 2nd harmonic.
Are you aware the phase of those harmonics are phsae locked with the fundamental? If you don't believe me, try squaring a sine wave. Or clip a waveform with two diodes and look at the result with an oscilloscope and question what you are seeing.
Attached is an LT spice simulation. Run the sumulation and press ctrl L at the end. It pops up a window showing you the results of an FFT. Just ignore the 4...9 harmonic because this is just noise.
Here is the FFT for the non linear transform:
Harmonic Frequency Fourier Normalized Phase Normalized
Number [Hz] Component Component [degree] Phase [deg]
1 1.000e+03 1.150e+00 1.000e+00 90.00° 0.00°
2 2.000e+03 1.998e-01 1.738e-01 180.00° 90.00°
3 3.000e+03 4.989e-02 4.340e-02 -90.02° -180.02°
And here is the FFT for the diode clipped waveform:
Harmonic Frequency Fourier Normalized Phase Normalized
Number [Hz] Component Component [degree] Phase [deg]
1 1.000e+03 7.833e-01 1.000e+00 90.00° 0.00°
2 2.000e+03 1.956e-05 2.496e-05 100.90° 10.91°
3 3.000e+03 1.025e-01 1.308e-01 90.01° 0.01°
No matter how many positive/negative supplies a power amp has or not has or how many times you invert the phases with transformers, the phase relation between the signals stays the same. Imagine what would happen if this would not be the case.