© 2020 IOP Publishing Ltd. The conventional semi-active variable damping (VD) device can only dissipate the vibration energy, which inherently limits its performance in vibration control. This paper proposes a novel semi-active inerter concept and develops a prototype with magnetorheological (MR) dampers, which can store the vibration energy in a flywheel and then release it to suppress vibration. The new idea is inspired by a variable inertance (VI) device, which has a VD device and a passive inerter in serial, and a passive mechanical motion rectifier (MMR), which can convert the reciprocating input into a unidirectional output. The controllable MMR (CMMR) applies two VD devices to replace the two one-way roller clutches in the MMR. By equipping the CMMR, the semi-active inerter gets more controllability than the original VI device because it can switch the torque and motion transmitting routes between the device terminals and the flywheel in it. The frequency-domain analysis validates the versatile of the semi-active inerter, which can work in VD and VI modes. The test results of the semi-active MR inerter prototype are used to identify the device parameters, which are applied for the control simulation. The semi-active MR inerter in CMMR mode has the best torque tracking performance with a given test condition, and it has a 20.13% improvement than in the VD mode. Then, a seat suspension with the semi-active MR inerter is applied to validate the effectiveness of the device in vibration control. The results show that the vibration reduction of the seat suspension in the CMMR mode is 39.3% higher than in the VD mode, which indicates a significant improvement of ride comfort. The new concept of the semi-active device has excellent potential in vibration control and is promising in practical applications.