Metempsychosis
Metempsychosis is a doctrine among some followers of Eastern teachings which expresses the theory of transmigration, that the human spirit may incarnate from one body to another, either human, animal, or inanimate, which is very different from the doctrine of reincarnation, which holds that man is an evolving being progressing through repeated human embodiments.
Controversy
René Guénon dedicates a whole chapter of his 1923 book The Spiritist Fallacy to the concept of Reincarnation. According to him, there is no such concept in any traditional religion. He claims that reincarnation is instead a modern concept created by spiritist and theosophist authors, mainly Allan Kardec, apparently inspired by a fatally flawed understanding of the traditional concepts of transmigration and metempsychosis.
Some may argue that transmigration of human souls into non-human bodies is implied in totemism.
Metempsychosis vs. Reincarnation
There is no authority in any of the sacred writings of Eastern religions for such a belief as transmigration. The only semblance to such an idea is found in the Kathopanishad (Chapter 5, Verse 9) which says that some of the souls, according to their deeds, return to the womb to be reborn, but others go into "the motionless", which is interpreted by some that they may reincarnate down even as low as the minerals. The Sanskrit word used in that place is "Sthanu", which also means "a pillar" and it is similiar to the passage in Revelation that says: "Him that overcometh, I will make a pillar in the house of my God, thence he shall no more go out." This passage is regarded in Esoteric Christianity with the meaning that when humanity has reached perfection, there will come a time when they will not more be tied to the wheel of births and deaths: that is, liberated from the "Rebirth" cycle (or "motionless").