If we have four grounded nodes in a circuit, can we connect two of them together and forget that those two were grounds?
I personally think the answer is no (...)


What you are thinking is correct. In order to have an equivalent circuit, you will need to connect all ground nodes together. Just 2 out of 4 will not work because you are changing Kirchoff's Current Law in the ground node, and so influencing the solution of the circuit.

Only if you know that it won't influence Kirchoff's Current Law, you can remove the ground reference. For example:







simulate this circuit &ndash; Schematic created using CircuitLab

Due to symmetry reasons, you can already see by inspection that the middle voltage is always 0V, so the GND-connection is not really necessary.

But it is not a general property.