Switchable crossover tool with hydraulic transmission

Switchable cross-over systems, devices, and methods for cementing well walls are provided. A switchable cross-over device includes a tool body and a flow sleeve. 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 flow sleeve is within the auxiliary chamber and movable between conventional and reverse circulation positions. In the conventional circulation position, the uphole and downhole tool paths are in fluid communication and the uphole annular port is in fluid communication with the downhole annular port through the auxiliary chamber. In the reverse circulation position, the flow sleeve forms first and second auxiliary flow paths in the auxiliary chamber, the uphole tool path and the downhole annular port are in fluid communication via the first auxiliary flow path, and the downhole tool path is in fluid communication with the uphole annular port.