Yes, it would ground the signal. Why do I prefer it? In short, it gave me the biggest single audio improvement in an amplifier I have ever experienced. And it was a direct A/B. About 15 years ago I was working on a small, single supply chip amp which used a LM3875 (now discontinued) in a differential topology. The preamp part was 4 op amps whose functions were, Differential Input, Volume Control, LP filter and Output, and then Inverter (and output). Though there was always a place for the 4th op amp, when I built the prototype I omitted the 4th amp as I already had far more gain than I needed and just used a second 27R resistor. This seemed to be confirmed as a good decision when the volume control (which I wouldn't do the same way now) ended up with a lovely taper that felt very classy. (It was a linear taper pot with a resistor in parallel to mimic a log taper and it seemed I had chosen the right values.) Anyway, it sat that way for about 18 months until one night I decided on a whim to put the final opamp in, to squeeze out the last bit of S/N. As I have said, the difference was quite stunning.
I don't think the improvement could have been just S/N, though 6dB is substantial. There could have been some vagaries in the chip but most likely is that it overcame some deficiency in the grounding scheme. That alone is a good reason to do it, especially as grounding and layout must be at the top of the list of perfectly good circuits failing to perform sonically. (Or, if you like, failing to meet distortion specs.) On this particular amp, where pre and power shared an SMPS, though each part had its own regulation inside the amp, a pointer might be that I remember having problems getting the sometimes very expensive pre amp regulators to perform to spec. So it's not just about CMR, but getting the right reference to its destination, which I suspect I failed to do across just 5" of wire!
Having a signal on the Cold could improve the probablity of success. I'd have to think this through more carefully, but maybe if the inversion happens in the second half of a dual op amp, you've virtually guaranteed it.
So, that's why I prefer it. Mainly my personal experience. But some of the stuff I'm working on at the moment actually needs a signal as the two halves have separate paths right through the box and don't meet anywhere to do their CMR thing or any distortion cancellation. Besides, who in audio electronics doesn't love symmetry?
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