This paper proposes a novel feeding network that enables the antenna array to perform dual-band continuous beam scanning and beam switching between forward and backward directions. Specifically, the switchable beam is controlled by the network input and the beam in each direction is continuously steerable at either frequency. To support a wide range of beam-scanning angles, a novel dual-band coupler with synchronously tuned differential phases is applied to the feeding network, together with a pair of phase shifters. The operating principles and design equations of the dual-band beam-scanning/switching network, coupler, and phase shifters will be given in detail. As a proof-of-concept, a 0.9-GHz/1.8-GHz beam-scanning/switching network was prototyped alongside a $1\times 4$ linear array. Experimental results show that the array can scan and switch its main beam from −24° to −42° in the backward direction and from 24° to 34° in the forward direction at 0.9 GHz, while the beam scanning is from −12° to −24° and from 12° to 22° at 1.8 GHz.