Real-time remote control over wireless is an important yet challenging application in fifth-generation and beyond due to its mission-critical nature under limited communication resources. Current solutions hinge on not only utilizing ultrareliable and low-latency communication (URLLC) links but also predicting future states, which may consume enormous communication resources and struggle with a short prediction time horizon. To fill this void, in this article we propose a novel two-way Koopman autoencoder (AE) approach wherein: 1) a sensing Koopman AE learns to understand the temporal state dynamics and predicts missing packets from a sensor to its remote controller and 2) a controlling Koopman AE learns to understand the temporal action dynamics and predicts missing packets from the controller to an actuator co-located with the sensor. Specifically, each Koopman AE aims to learn the Koopman operator in the hidden layers while the encoder of the AE aims to project the nonlinear dynamics onto a lifted subspace, which is reverted into the original nonlinear dynamics by the decoder of the AE. The Koopman operator describes the linearized temporal dynamics, enabling long-term future prediction and coping with missing packets and closed-form optimal control in the lifted subspace. Simulation results corroborate that the proposed approach achieves a lower mean squared control error at 0-dBm signal-to-noise ratio (SNR) than the nonpredictive baseline.