Understanding Rgain of an instrumental amplifier

I answered this SE EE question a few hours ago.

My answer

The idea

The role of the common resistor Rgain is essential to the circuit operation. Connected in this way, the input amplifier stages can interact with each other and behave differently depending on the mode. The voltage of the midpoint inside the resistor Rgain indicates the mode. So it is important to monitor it and this is achieved by "split" Rgain in two or "assemble" it from two resistors Rgain/2 in series.

Operation

In the common mode, the midpoint follows the variations of the two input voltages. Figuratively speaking, it "moves" with them and acts as a "movable ground". The two input stages cooperate and act as voltage followers; so the input signal is not amplified.

In the differential mode, the midpoint "freezes" and acts as а "fixed" or "virtual" ground. The two stages counteract each other and function as non-inverting amllifiers; so the input signal is amplified.

Conclusion

So, looking for the most essential in this beautiful symmetrical circuit solution, we can find it in the dual behavior of the midpoint inside the resistor Rgain: on the one hand, in common mode, it follows the input voltages; on the other hand, in differential mode, it has a fixed voltage. Figuratively speaking, in common mode it is "soft" and "movable" while in differential mode, it is "stiff" and "immovable".

See more about the philosophy behind this ingenious circuit solution in my Codidact paper What is the idea behind the op-amp instrumentation amplifier?