: We know the soldier Orcs had identification numbers. And we know there were hordes of slaves in Mordor, some human, some Orcish, which performed non-military tasks, such as raising food, caring for the roads, etc.

: Minas Morgul was "ruled" by the Lord of the Nazgul, and Dol Guldur was "ruled" by Khamul the Easterling. These were both on a par with "colonies" or "satellite states", I think, even though Dol Guldur was technically more ancient than Mordor (in the Third Age).

: The Mouth of Sauron was the Lieutenant of Barad-dur, and Gothmog was the Lieutenant of Minas Morgul. I suspect these were "ranks", not in the sense of "lieutenant answers to captain", etc., but rather, in the sense that they indicated special positions of importance within the hierarchy.

: So, I'm curious, how would the rest of you devise a structure for representing Mordor's government and military?

The rule of fear, the ultimate expression of evolution - only the fittest and strongest survive.
Promotion based on strength through a similar 'army' structure and contorl of birth to the extent of being 'born into' the army, a slave, a builder etc etc.
'Deocracy' , I think the word you want is 'Theocracy' - government by God or a god
Dave