“In Eastern Orthodox christianity the Emperor dominated the Patriarchy and exercised jurisdiction in both roles under what is called Caesaro-papism.”

More true of the earlier periods. During the Iconoclast controversy the church actually put up a heck of a fight against the Emperor(s), and while the controversy was actually settled when the Emperess Irene switched the Imperial position, the Orthodox church actually gained a great deal of independence. After that point you do have instances of the Patriarchy standing up to the Emperor (Nicophorus the something was excommunicated for marrying a fourth time, IIRC).