How is this 2-phase system wired into panels? Is there a 2-phase panel that accepts 5 conductors for the feed?
If I understand what you said correctly, you have 4 "hots" to choose from to use with the neutral, but the hot to hot connection can only be between the phases from the same source?
I can't tell you, not because it is classified but I just don't know
This kind of source is probably older than both of us put together, and maybe even some more. I don't really know how it was installed, but was more common before modern "panelboards" were around for certain. I do understand the principle of how they work, but don't know how distribution was handled.
Draw yourself two sets of winding coils with centertaps that cross one another perpendicular at both centertaps. That is your basic representation of how the source is connected. Voltage and current in each winding is out of phase from the other winding, the windings of a two phase motor are connected similarly so with the two resulting currents being out of phase we get rotation of their magnetic fields to drive the motor.
Your single phase induction motors are two phase motors, we just manufacture a second phase with things like capacitors, different reactance values in two different coils, etc. A real two phase motor just happens to have two voltage/currents input that are out of phase with one another.