Even loading of both supply rails... by inverting one channel

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Since quite a bit of the same sound exist in both channels especially bass, both channels more or less draw similar amount of current from the positive or negative rails at the same time.

What if, one channel is inverted so it outputs negative while the unaltered channel outputs positive, so both rails are more evenly loaded. The inverted output isn't an issue with speakers where all that's needed is some rewiring.

This could improve performance slightly and be a bit easier on virtual ground circuits, but I haven't seen this being done for some reason?
 
A very old idea. At the time of the NAD3020 a popular tweak was to invert one channel of the phono cartridge and the corresponding speaker. This changed the sound quite substantially as the power supply was indeed marginal. Don't mean to say the sound actually became better, just different, with a wider sound stage and a whole in the center.
 
This is also often done in Class-D amplifiers with single-enden output stages. These are known to suffer from rail-voltage pumping effects while energy is going back to the power-supply bulk caps and therefore will rise the rail voltages - sometimes to the maximum voltage level of the driving chips or the switching transistors. Definately something you don't want if you consider reliability of the design.
With bridged Class-D amplifier this will not happen because the energy will not go back to the rails but (through the rails) to the other inverted driving output - completely avoiding the rail voltage pumping effect.
This effect, by the way, will be worse with mono signals being present in the audio signal. And this is usually the low frequency region with the high power bass signals compared to the mid or high frequency region.
Similar is the effect on analog power amplifiers, except that the rail voltage pumping effect does not happen.
If you compare the scheme for a bridged amplifier with your modified stereo amplifier you'll see that this is exactly the same hardware configuration. One channel is inverted and the output of the inverted channel goes to the (-) pole of the speaker. The only difference is the usage of two speakers instead of one for the bridged mode.
 
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