Switchable crossover tools can include a tool body and a rotatable chamber. The tool body includes a main tool path separable into uphole and downhole tool paths and an auxiliary chamber containing uphole and downhole annular ports. The rotatable chamber is located and rotatable within the auxiliary chamber and forming first and second auxiliary flow paths through the auxiliary chamber. The rotatable chamber is positionable between a conventional circulation mode and a reverse circulation mode. In the conventional circulation mode, the uphole and downhole tool paths are in fluid communication and the uphole and downhole annular ports are in fluid communication through the auxiliary chamber. In the reverse circulation mode, the uphole tool path is in fluid communication with the downhole annular port via the first auxiliary flow path, and the downhole tool path is in fluid communication with the uphole annular port via the second auxiliary flow path.