Rejecting the idea that mental phenomena can cause physical phenomena, makes it possible to accept both other premises:
Mental phenomena are not identical to physical phenomena.
Mental phenomena are causally effective in the area of physical phenomena.
Interactionist dualism has a long tradition in philosophy. Probably the most popular version of this position was developed by Descartes in the 17th century. In his conception two kinds of substances exits: the one of the mind (res cogitans) and an extended one (res extensa). Causation in this conception occurs in both directions. Answering the question of Elizabeth of Bohemia how the human soul by being only a thinking substance can have effects on the physical world, as it can be seen in someone’s performing a voluntary action, Descartes was introducing the human epiphysis to act as a miraculous intermediary between the two substances. A recent interactionist approach relocates this interplay to the level of quantum mechanics. Anyway, not interactionalist theory to my knowledge ever proposed a convincing hypothesis on the nature of the very interaction between mental and physical states.