Hinduism supports reincarnation as well as Buddhism.
Technically speaking, this isn’t completely accurate because Buddhism teaches ”rebirth’ rather that reincarnation.
Reincarnation in Hinduism is the transfer of the ”soul’ (atman) from one being/creature/plant to another. Whereas Buddhism promotes there being no ”soul’ (anatman), and that each ”incarnation’ is in fact only a rebirth of the same entity with nothing transferring between each ”incarnation’.
Basic explanation can be found
here
So the term most often used is rebirth, rather than reincarnation. Reincarnation implies the transfer of an essence, or a soul, while rebirth follows the law of causality, or dependant origination, where this arises because of circumstances which happened before.
Brian