Fundamental relationships are believed to exist between the symmetries of building blocks and the condensed matter phases that they form(1). For example, constituent molecular and colloidal rods and disks impart their uniaxial symmetry onto nematic liquid crystals, such as those used in displays(1,2). Low-symmetry organizations could form in mixtures of rods and disks(3-5), but entropy tends to phase-separate them at the molecular and colloidal scales, whereas strong elasticity-mediated interactions drive the formation of chains and crystals in nematic colloids(6-11). To have a structure with few or no symmetry operations apart from trivial ones has so far been demonstrated to be a property of solids alone(1), but not of their fully fluid condensed matter counterparts, even though such symmetries have been considered theoretically(12-15) and observed in magnetic colloids(16). Here we show that dispersing highly anisotropic charged colloidal disks in a nematic host composed of molecular rods provides a platform for observing many low-symmetry phases. Depending on the temperature, concentration and surface charge of the disks, we find nematic, smectic and columnar organizations with symmetries ranging from uniaxial(1,2) to orthorhombic(17-21) and monoclinic(12-15). With increasing temperature, we observe unusual transitions from less- to more-ordered states and re-entrant(22) phases. Most importantly, we demonstrate the presence of reconfigurable monoclinic colloidal nematic order, as well as the possibility of thermal and magnetic control of low-symmetry self-assembly(2,23,24). Our experimental findings are supported by theoretical modelling of the colloidal interactions between disks in the nematic host and may provide a route towards realizing many low-symmetry condensed matter phases in systems with building blocks of dissimilar shapes and sizes, as well as their technological applications. Dispersion of colloidal disks in a nematic liquid crystal reveals several low-symmetry phases, including monoclinic colloidal nematic order, with interchange between them achieved through variations in temperature, concentration and surface charge.