In this study, we presented the design and testing of a microwave photonic filter with switchable single- and dual-passband operation based on stimulated Brillouin scattering. By adjusting the interval between the wavelengths of the pump light and the signal light, the filter can be switched from single- to dual-passband operation. Moreover, the passband frequency can be tuned. In addition, we prepared a theoretical model of this device and made the corresponding simulation. The analysis was verified using a proof-of-concept experiment. The experimental results show that the rejection ratio can be up to 30 dB, and the frequency tuning range is approximately 0-18 GHz. The experimental results are observed to agree well with the simulated results.