You just about have your answer right in the question. The short answer is that moving eastward seems to relate to exile, while moving westward is a return to the garden and the presence of God.
The long answer:
The garden is planted in the east of Eden
The garden is the primeval meeting place between God and man. It is the first sanctuary, where man is placed to work (serve) and keep (guard) (Gen 2:15). These are the same tasks of the priests and Levites with regard to the tabernacle (cf. Num 18:1-7).
Cherubim are stationed on the east side of the Garden of Eden
When Adam and Eve are exiled from the presence of God and from access to the tree of life, they are sent out eastward. To re-enter the garden, they would have to travel west and pass the cherubim and flaming sword. Also, after Cain murders his brother, he is driven even further east, further from the garden. By implication, the cherubim are also guarding the way into Eden itself, which is higher up and further west than the garden. I'll come back to this.
Parts of the burnt offering are to be thrown to the east side of the altar
This is part of the fowl Ascension ritual. It is the 'crop', which is somehow associated with the tail feathers. So, the rear of the bird is thrown to the east, while the head and body move westward, into the fiery presence of God.
The tabernacle's entrance faces east
This implies that the tabernacle is a new, more glorified garden sanctuary. Armed priests and Levites guard the tabernacle (Num 18:4) just like the cherubim who guarded the entrance of the garden. They also wield the flaming sword. In order for a worshiper to draw near to God, he would have to pass through the knife and the fire. He can't do this without dying, so he leans his hand on the head of an animal to commission it to go as his representative (cf. Num 8:10-18, 27:18-23). The animal passes through the knife and fire, moving westward and upward into the presence of God (it is "burned up" Lev 1:13 - lit. transformed into smoke, thus joining the glory cloud over the tabernacle)
In Ezekiel's vision God's glory comes from the east and enters the temple from the east.
Earlier (ch 10), we see the glory of God depart from the temple, traveling eastward to join the exiles in Babylon. Here, in chapter 43, Ezekiel sees a restored, more glorious temple, and the glory of God moves back in.
The same temple faces east with a river flowing east from it
This goes all the way back to the garden. A river flowed out of Eden - that is, from higher up and further west - through the garden and then out to the outlying lands. This seems to imply that moving west into the land of Eden would be desirable and would be moving closer to the throne of God (at least symbolically). Solomon's temple was also a garden sanctuary, with carvings of trees and fruit on the walls (cf. 1 Kings 6). It had a great bronze sea and bronze water chariots, which lined the north and south sides of the courtyard (1 Kings 7:23-39). The chariots with wheels imply a flow of water and correspond to the river that flowed through the garden. This is expanded further in Ezekiel's vision in chapter 47 to an increasingly deep and wide river coming from the temple and flowing to the outlying lands. This is picked up again in Revelation 22, where an angel shows John a vision of the New Jerusalem, the Church, the City-Bride of Christ, and "the river of the water of life, bright as crystal, flowing from the throne of God and of the Lamb through the middle of the street of the city; also, on either side of the river, the tree of life with its twelve kinds of fruit, yielding its fruit each month. The leaves of the tree were for the healing of the nations."
