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Switchable metamaterial absorbers/reflectors (MAs/MRs) are important bifunctional electromagnetic devices and have been the subject of numerous scientific studies. However, there is a lack of bifunctional devices that operate in the terahertz band. Here, we theoretically propose a broadband switchable MA with many excellent properties, such as good thermal stability, high insensitivity to inferior film quality of the graphene, excitation polarization and wide incident angles, and outstanding structural parameter tolerance. The bandwidth of the proposed broadband MA is 3.4 THz with an absorptivity over 90% in the frequency band of 1.6-5 THz. The proposed absorber can switch to a reflector with a reflectivity over 93% by tuning the chemical potential of the graphene and reducing the temperature. Therefore, the switching intensity of the proposed MA exceeds 83%. The physical mechanisms of the broadband absorption of the proposed structure are investigated using the impedance matching theory and the multiple reflection interference theory. The reflection mechanism of the proposed broadband reflector is discussed by analyzing the effective parameters. The absorption and switching mechanism are theoretically investigated by performing detailed numerical calculations to analyze the surface loss intensity, electric field, and magnetic field. These findings can accelerate the development of terahertz broadband switchable devices.