We theoretically prove in this paper that using planar multilayer graphene-hexagonal boron nitride (hBN) metamaterials (GhMMs) can yield ultrabroadband and high-contrast asymmetric transmission (AT) and asymmetric absorption (AA) of light. The AA and AT features are obtained in the far-infrared (FIR) and mid-infrared (MIR) regions for normally incident light with transverse magnetic polarization. Here, the GhMMs are integrated with two asymmetric gratings of Ge and are composed of alternating multilayers of graphene (11 multilayers) and hBN layers (10 layers). Moreover, the total subwavelength thickness of the hybrid structures is about 3 μm, being less than half of the free-space wavelength up to nearly 50 THz. This approach-which is similar to the one introduced by Xu and Lezec [Nat. Commun. 5, 4141 (2014)] for a passive hyperbolic metamaterial operating in the visible range-is based on the excitation of high-β modes of the GhMM with different transmission characteristics. In addition to being ultrabroadband and high-contrast, AT and AA features of the proposed GhMMs can be actively tuned by varying the chemical potential of graphene. Furthermore, it is shown that an onoff switching of AT factor at FIR and selective tunability at MIR frequencies can be obtained via varying μ. Due to its subwavelength and planar configuration and active operation, these multilayer graphene-hBN metamaterials with AT and AA characteristics hold promise for integration with compact optical systems operating in the MIR and FIR ranges and are suitable for applications such as optical diodes, sensors, and thermal emitters.