A crossover tool has an internal sleeve rotatably positioned within an external sleeve, and each of the sleeves has ports alignable with ports on the other sleeve. After deploying the crossover tool downhole and diverting fluid flow below the tool, fluid flow communicated into the internal sleeve tends to rotate it relative to the external sleeve until the ports are substantially aligned so that wear to the components is substantially reduced. The ports themselves may facilitate the rotation and alignment. For example, ports on the internal sleeve may produce tangentially exiting fluid flow. Alternatively, an additional outlet may be defined in the internal sleeve and eccentrically located to its rotation axis. Furthermore, an internal sleeve or insert may partially block fluid flow through the ports to allow greater fluid flow through the additional outlet to enhance rotation of the internal sleeve.