We report magnetometry data obtained on twin-free single crystals of , which is considered a candidate material for realizing the Kitaev honeycomb model for quantum spin liquids. Contrary to a common belief that such materials can be modeled with the symmetries of an ideal honeycomb lattice, our data reveal a pronounced twofold symmetry and in-plane anisotropy of over 200%, despite the honeycomb layer’s tiny orthorhombic distortion of less than 0.2%. We further use magnetic neutron diffraction to elucidate a rich variety of field-induced phases observed in the magnetometry. These phases manifest themselves in the paramagnetic state as diffuse scattering signals associated with competing ferromagnetic and antiferromagnetic instabilities, consistent with a theory that also predicts a quantum spin liquid phase nearby. Our results call for theoretical understanding of the observed in-plane anisotropy and render a promising ground for finding exotic quantum phases by targeted external tuning.