Transmission systems that enable speed, torque, and direction conversion with high responsiveness and back-drivability are strongly required for higher performance robotic systems. Engagement states can be changed by clutches, but traditional clutches cannot change the direction and provide sufficient backdrivability. On the other hand, direction conversion can be realized by gear box such as bevel gears, but its backdrivability is poor. Thus, we newly develop a prototype transmission system with backdrivablity and responsiveness integrating clutch and direction converter using magnetorheological fluid (MRF). MRF is a functional fluid consisted of magnetic particles and carrier fluids which can change its viscosity rapidly and continuously according to the strength of magnetic field. MRF clutch consists of driving and driven shaft connected by vanes and coils for controlling MRF. MRF direction converter consists of bevel gears, brake, and MRF for reversing the output direction. In addition to traditional functions, such as speed and torque conversion, the proposed power transmission system provides five working modes: forward direction, reverse direction, and three kinds of free, according to states of the MRF clutch, MRF direction converter, and brake. Preliminary experiments revealed that the proposed transmission system could adequately realize implemented functions.