We consider the varieties and directions of influence that the source and target domains involved in a conceptual metaphor can have on each other during the course of understanding metaphorical utterances based on the metaphor. Previous studies have been restricted as to direction of influence and as to type of influence. They have been largely confined to the "forward" (source-to-target) direction of influence, and they have concentrated on the transfer of features or propositions and (to some extent) the highlighting of aspects of a domain. By contrast, this article stresses the importance of other varieties of influence (e.g., transfer of queries and uncertainty effects) and of "reverse" influence (target-to-source). We seek to curb the natural tendency to think that because metaphor involves an overall move from source to target, therefore all interdomain influences in the course of understanding go in that direction. The bulk of the article explores the theoretical issues involved. These issues have arisen out of a reasoning-based approach to metaphor, seeking to make best use of information from metaphorical utterance, context, source and target domains, and known metaphorical mappings by applying a complex overall reasoning process. The article briefly explains the thorough implementation of reverse influence in a computer program for metaphorical reasoning called ATT-Meta, although the theoretical considerations are relevant to all disciplines concerned with metaphor. We make some suggestions for further computational and psychological research on metaphor.