Either one is a good choice for an entity's symbol. Having French roots, I have seen the fleur de lis on plenty of items. It is crisp and clean. There is a simple beauty to it.
I am also a believer that the fleur de lis was a stylized homage tot he double headed eagle. The DHE is much more fierce looking and signifies the power of the monarch. The might as well as legal leadership is symbolized by the eagle's talons gripping a monarch's symbols. The two heads can look east and west to always have an eye on the borders and project the face of the empire to those barbarians. It is used quite often and very well by monarchies through the years.