  Chiral symmetry breaking is ubiquitous in biological systems, from DNA to
bacterial suspensions. A key unresolved problem is how chiral structures may
spontaneously emerge from achiral interactions. We study a simple model of
bacterial suspensions in three dimensions that effectively incorporates active
motion and hydrodynamic interactions. We perform large-scale molecular dynamics
simulations (up to $10^6$ particles) and describe stable (or long-lived
metastable) collective states that exhibit chiral organization although the
interactions are achiral. We elucidate under which conditions these chiral
states will emerge and grow to large scales. We also study a related
equilibrium model that clarifies the role of orientational fluctuations.
