Yes, all of the Eastern Catholic Rites are in communion with Rome, and together with the Latin Rite, they make up the Catholic Church. That’s why they’re called Eastern Catholic Rites. Members of the Latin Rite Roman Catholic Church do fulfill their Sunday obligation if attending the Sunday worship of an Eastern Catholic Church, and they are welcome to receive Holy Communion.

There are 22 Eastern Rite Catholic Churches, and they follow different Eastern Christian liturgical traditions: Alexandrian, Antiochian, Armenian, Byzantine, and East Syrian Rite. You’ll find a full listing of the 22 particular churches and more information about them here.