Over the last few years, broken symmetry within crystals has attracted extensive attention since it can improve the control of light propagation. In particular, low-symmetry Bravais crystal can support shear polaritons, which has great potential in thermal photonics. In this work, we first use the fluctuation-dissipation theorem to investigate mechanisms of near-field thermal radiation (NFTR) in a low-symmetry Bravais crystal. The NFTR between such crystal slabs is nearly four orders of magnitude larger than the blackbody limit, demonstrating its remarkable potential for noncontact heat dissipation for nanoscale circuits or other devices. Moreover, we report a form of twist-induced near-field thermal control system employing the low-symmetry Bravais crystal medium (β-Ga2O3), showing that this crystal can serve as an excellent platform for twist-induced near-field thermal control. Due to the intrinsic shear effect, the twist-induced modulation supported by low-symmetry Bravais crystal exceeds that by high-symmetry crystal. We further clarify how the shear effect affects the twist-induced thermal-radiation modulation supported by hyperbolic and elliptical polaritons and show that the shear effect significantly enhances the twist-induced thermal control induced by the elliptical polariton mode. These results open directions for thermal-radiation control in low-symmetry materials, including geological minerals, common oxides, and organic crystals.