Bioconvection is the prototypical active matter system for hydrodynamic instabilities and pattern formation in suspensions of biased swimming microorganisms, particularly at the dilute end of the concentration spectrum where direct cell–cell interactions are less relevant. Confinement is an inherent characteristic of such systems, including those that are naturally occurring or industrially exploited, so it is important to understand the impact of boundaries on the hydrodynamic instabilities. Despite recent interest in this area, we note that commonly adopted symmetry assumptions in the literature, such as for a vertical channel or pipe, are uncorroborated and potentially unjustified. Therefore, by employing a combination of analytical and numerical techniques, we investigate whether confinement itself can drive asymmetric plume formation in a suspension of bottom-heavy swimming microorganisms (gyrotactic cells). For a class of solutions in a vertical channel, we establish the existence of a first integral of motion, and reveal that asymptotic asymmetry is plausible. Furthermore, numerical simulations from both Lagrangian and Eulerian perspectives demonstrate with remarkable agreement that asymmetric solutions can indeed be more stable than symmetric; asymmetric solutions are, in fact, dominant for a large, practically important region of parameter space. In addition, we verify the presence of blip and varicose instabilities for an experimentally accessible parameter range. Finally, we extend our study to a vertical Hele-Shaw geometry to explore whether a simple linear drag approximation can be justified. We find that although two-dimensional bioconvective structures and associated bulk properties have some similarities with experimental observations, approximating near-wall physics in even the simplest confined systems remains challenging.