Do any Catholic Liturgical Rite follow the Ad Orientem form of Holy Mass?
The short answer is yes.
The tradition of ad orientem worship is apostolic in origin. But despite that fact, there are more than a few Latin Catholics who are wary of such a tradition. This practice is still permitted in the Ordinary Form of the Mass. This in fact was not changed by Vatican II.
Obviously, the Extraordinary Form of the Mass is still celebrated ad orientem*. Additionally, ordinariates created under Anglicanorum coetibus celebrate the Anglican Use Mass ad orientem.
Many of the Eastern Catholic Churches, specifically those of the Byzantine Rite, exclusively worship ad orientem.
When you hear certain Latin Catholics express their dislike of traditional liturgical staples like ad orientem worship, or sacred chant, elaborate vestments, and the use of incense, all things that constitute the norm in the Byzantine Rite, how does it make you feel when your own fellow Catholics say these things? - A Byzantine Look at Worshipping Ad Orientem
The Code of Canons of the Eastern Churches actually asks theses Catholic Eastern Rites to preserve this tradition.
Instruction for Applying the Liturgical Prescriptions of the Code of Canons of the Eastern Churches
- Prayer facing the east
"Ever since ancient times, it has been customary in the prayer of the Eastern Churches to prostrate oneself to the ground, turning toward the east; the buildings themselves were constructed such that the altar would face the east. Saint John of Damascus explains the meaning of this tradition: "It is not for simplicity nor by chance that we pray turned toward the regions of the east (...). Since God is intelligible light (1 Jn. 1:5), and in the Scripture, Christ is called the Sun of justice (Mal. 3:20) and the East (Zec. 3:8 of the LXX), it is necessary to dedicate the east to him in order to render him worship. The Scripture says: 'Then the Lord God planted a garden in Eden, in the east, and he placed there the man whom he had formed' (Gen. 2:8). (...) In search of the ancient homeland and tending toward it, we worship God. Even the tent of Moses had its curtain veil and propitiatory facing the east. And the tribe of Judah, in as much as it was the most notable, encamped on the east side (cf. Nm. 2:3). In the temple of Solomon, the Lord's gate was facing the east (cf. Ez. 44:1). Finally, the Lord placed on the cross looked toward the west, and so we prostrate ourselves in his direction, facing him. When he ascended to heaven, he was raised toward the east, and thus his disciples adored him, and thus he will return, in the same way as they saw him go to heaven (cf. Acts 1:11), as the Lord himself said: 'For just as lightning comes from the east and is seen as far as the west, so will the coming of the Son of Man be' (Mt. 24:27). Waiting for him, we prostrate ourselves toward the east. It is an unwritten tradition, deriving from the Apostles."
"This rich and fascinating interpretation also explains the reason for which the celebrant who presides in the liturgical celebration prays facing the east, just as the people who participate. It is not a question, as is often claimed, of presiding over the celebration with the back turned to the people, but rather of guiding the people in pilgrimage toward the Kingdom, invoked in prayer until the return of the Lord."
Cardinal Joseph Ratzinger, who later became Pope Benedict XVI, described the eastward orientation as linked with the "cosmic sign of the rising sun which symbolizes the universality of God." He also states in the same book (The Spirit of the Liturgy) that:
Despite all the variations in practice that have taken place far into the second millennium, one thing has remained clear for the whole of Christendom: praying toward the east is a tradition that goes back to the beginning. Moreover, it is a fundamental expression of the Christian synthesis of cosmos and history, of being rooted in the once-for-all events again.
Syriac and Arabic Christian apologetics of the 7th century explained that Christians prayed facing east because "the Garden of Eden was planted in the east (Genesis 2:8) and that at the end of time, at the second coming, the Messiah would approach Jerusalem from the east." Saint John Damascus taught that believers pray facing east because it "reminds Christians of their need to long for and strive for the paradise that God intended for them" and because "Christians affirm their faith in Christ as the Light of the world" by praying in the direction of sunrise.