Joseph Atwill in "Caesar's Messiah" makes the case that the Caesars were not only kings but high priests over the leading cults. Separation of church and state is a relatively new concept.
Herod the Great sought to control Judaism in the same way. Through his Hasmonean wife he intended to produce offspring that were considered qualified as high priests in the temple of Jerusalem. We should also expect that the Herodians demanded to be recognized as the leaders of other Jewish sects and to the maximum extent that those sects would tolerate. This might have even entailed assuming false or double identities.
Temple cults outside Israel seem to have been far more amenable to Roman dominance. Herod and his family had to be much more subtle within their own dominions.
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