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In this paper, we show a new bandstop filter circuit topology. Unlike conventional bandstop filter circuit topologies, the new circuit topology has inter-resonator coupling structures. The presence of these inter-resonator coupling structures enables convenient switching from a bandstop to a bandpass filter. Using the new bandstop filter topology, this paper demonstrates a design of a frequency-agile bandstop-to-bandpass switchable filter. The filter is composed of tunable substrate-integrated cavity resonators and can be switched to have either a bandstop or bandpass response. Switching is achieved by turning on and off switches placed within the filter structure. A prototype of the proposed design is fabricated and the concept is verified experimentally.