Well there are times where we do use a center tapped transformer to “split phases”. We are looking to have, on one secondary winding, the original waveform, and on the other one the same exact waveform with polarity inversion.
There is not one original waveform. There are two original waveforms.
We certainly don’t want something displaced in time by 180 degrees. Doesn’t matter that this isn’t electrical distribution as transformers don’t know and obey the same laws regardless.
No one has said there is a time displacement. That requires a different type circuit. Both waveforms start at the same time.
You keep thinking the larger waveform defines a direction for the smaller waveforms. It does not. By that logic, 120/208 would have to be defined L2->N and N->L1. Then you would have 120<180d and 120<120d. Can't you see the problem with the inconsistent logic? Here is the problem:
For the 120/240 system you want to have 120<0d + 120<0d = 240<0d. You should recognize that you can also have 120<0d - 120<180d = 240<0d.
For the 120/208 system you want to have 120<0d - 120<120d = 208<-30d. You should recognize that you can also have 120<0d + 120<300d = 208<-30d.
Both methods are valid ways to take waveforms and is not about inverting an "original" but defining the original the way you are taking the waveforms to start with. There is no pre-determined "original".
The logic of using only one fails because if you recognize that you can take the difference waveforms for the 120/208 system then you should recognize you can take the difference waveforms for the 120/240 system. That is the way voltages work. You define the voltages. It is really that simple.