The formation of hybrid light-matter states through the resonant interaction of confined electromagnetic fields with matter excitations has emerged as a fascinating tool for controlling quantum-mechanical states and then manipulating the functionalities and chemical reactivity landscape of molecular materials. Here we report the first observation of switchable strong light-matter coupling involving bistable spin-crossover molecules. Spectroscopic measurements, supported by transfer-matrix and coupled-oscillator simulations, reveal Rabi splitting values of up to 550 meV, which at 15% of the molecular excitation energy enter the regime of ultrastrong coupling. We find that the thermally induced switching of molecules between their low-spin and high-spin states allows fine control of the light-matter hybridization strength, offering the appealing possibility of reversible switching between the ultrastrong- and weak-coupling regimes within a single photonic structure. Through this work, we show that spin-crossover molecular compounds constitute a promising class of active nanomaterials in the burgeoning context of tunable polaritonic devices and polaritonic chemistry.