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The crimson phoenix emblazoned banner

This is the official banner of The Red Company and it represents the fierce character of Rossiel, the ancestor of the current Dúnedain of the company. Rossiel was red-haired and although none of us are certain, it might be that the red company's name is a derivation of this characteristic. The personality of Rossiel may be compared to the phoenix. Phoenixes are very fiery and every time they die, they will be reborn out of their own ashes. So was Rossiel: she did not give up and she always knew to get new courage and to inspire her warriors. Even 2000 years later, she is not forgotten and inspires the red company still.

Source: 
Credits go to Nanteana of The Red Company!