"Nishiki no Mihata" is basically the flag of the Emperor. Flying it is essentially saying you're fighting the enemies of the Imperial Court. The standard image would be the gold sun flag over a red background, though it technically is a set, not just one. The other flag of the pair is the silver moon flag. It is also a word for referring your just cause. Nishiki is silk, by the way.
The sun representation is obvious since that's the Imperial family with Amaterasu, but I have no f#cking clue what the Tsukuyomi counterpart stands for beyond being a natural cosmic pair as the moon.
