The Officer-Batman relationship is an archetype in British literature, and culture in general, that is based on ranking in the British Army. The Officers each have a specially assigned recruit or operative who is essentially a side kick. This is the archetype that many relationships depicted in modern culture, such as Batman and Robin, or really any hero/sidekick pair, are based on. The relationship is one of respect, what with the Batman being lower in rank. But at the same time, it is of significant importance emotionally and personally in that the officer has many responsibilities, and the Batman has to be there to keep the Officer in good spirits through thick and thin.
Classic examples in modern entertainmen of the Officer-Batman relationship:
-Frodo and Sam in The Lord of the Rings Trilogy
-Lieutenant Dan and Forrest in Forrest Gump
-Harry and Ron in the Harry Potter series
-Tony and Manny in Scarface
-Herb Brooks and Craig Patrick in Miracle
I think this relationship exemplifies any dual relationship: two partners, equally acknowledging their strengths and weaknesses, and cooperating in accordance with them in order to acheive their goal in the best way possible. However, I think the nature of the Officer-Batman relationship in particular is one most like the ExTx - IxFx dual relationships: one performs the majority of the practical tasks, while the other offers emotional recharge. In particular, it seems to exemplify the two ENTx-ISFx couples: one does the majority of the big picture thinking and stratgizing while the other takes care of the "little things" and makes things run smoothly. With the emphasis on status based on position and the maintenance of the emotional aspect of the relationship being of particular importance, I would say that it fits the Gamma quadra best, and is probably best displayed by the archetype LIE-ESI dual pair.