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Curiosity got the best of me today, I have been aware of this switch for a long time. I would really like to understand it, and how it relates to the inverter. This switch is located behind a false wall in a closet directly above the incoming street power. (Hint #1)  

I am "assuming" that if no 120 volt street or generator power is available, this relay having no coil current, allows both 120 volt circuits to energize the two inverter circuits in the coach?

So unfortunately I am not able to tell you exactly how this is wired but I can give you some tips for possibly figuring it out, unless someone else has already done this. So the orange and green wires are connected to the relays coil. If you follow the orange wire it appears that it leaves the enclosure and i suspect this is the wire that is controlling the relay. You should also be able to look at the side of the relay and see what the coil voltage is, that will also give you a clue as to what is controlling this relay, 12v DC or 120v AC. So you could also use your meter to see where you have voltage and when. 
So if it works as if I think it does here is what I think you should find. 
The top two screws should go out to the load, ie the receptacles or whatever is on this circuit. 
The middle screws should be the incoming power from your power panel, ie park power or generator power.
The bottom screws would be coming from the inverter. 
So you can turn on your park power and see if you have power at the middle screws.
Turn on and off your inverter and see if it corresponds with the bottom screws.

So I suspect this probably works similar to how our Wanderlodges. We have the same kind of relay for our inverter circuit. The way ours works is that if you are on park power or generator power it energizes the coil and connects the circuit to the power panel. If there is no park power or generator power then the coil is not energized and the springs in the relay keep the contacts closed and its connected to the inverter, providing inverter power to whatever is connected to the relay. In our case we have electric drapes and a couple of outlets that are connected to this circuit.