The examination of Triply Periodic Minimal Surfaces (TPMSs) in the context of their acoustic dampening qualities has established them as viable alternatives to conventional porous absorbers. By adjusting the structure's porosity, one can significantly alter the TPMSs' acoustic absorption characteristics. Traditionally, enhancing sound absorption has involved decreasing porosity, which unfortunately increases the material's weight. In this study, we introduce a different approach to alter their absorption performance without altering their overall relative density. Our approach relies on leveraging the mathematical formulation of such structures to break their underlying geometrical symmetry. This symmetry-breaking results in altering their sound absorption characteristics while introducing an anisotropic acoustic response. In this paper, we present the mathematical design and fabricate the samples using the Fused Filament Fabrication (FFF) additive manufacturing method. The sound absorption characteristics of the printed samples are tested using a two-microphone normal incidence impedance tube setup. The presented results demonstrate that symmetry-breaking provides a viable route towards improving the low-frequency response of TPMS sound absorbers.