The two legions are both very focused on actual objectives, they're great at thinking outside the box and have a strong sense of foreseeing and planning for contingencies. Both are very independent and used to relying on nobody except their own legion brothers, so the cooperation would probably be very loose, as they wouldn't want others getting in the way of their plans.
However, they would have excellent coordination, even though communication would consist solely in relaying the important facts. While both are reluctant to place their trust in others, they are also pragmatic enough to cooperate efficiently without expecting trust or genuine friendship from their allies.
Neither have any particular ideologies they would enforce on others, and they rarely, if at all, fight a war for the sake of proving a point, of for the sake of war itself. Neither have any patience for pomp, ceremony or protocol, and couldn't give a damn about concepts of honor or faith.