ABSTRACT Magnetically arrested accretion discs (MADs) around black holes (BHs) have the potential to stimulate the production of powerful jets and account for recent ultra-high-resolution observations of BH environments. Their main properties are usually attributed to the accumulation of dynamically significant net magnetic (vertical) flux throughout the arrested region, which is then regulated by interchange instabilities. Here, we propose instead that it is mainly a dynamically important toroidal field – the result of dynamo action triggered by the significant but still relatively weak vertical field – that defines and regulates the properties of MADs. We suggest that rapid convection-like instabilities, involving interchange of toroidal flux tubes and operating concurrently with the magnetorotational instability (MRI), can regulate the structure of the disc and the escape of net flux. We generalize the convective stability criteria and disc structure equations to include the effects of a strong toroidal field and show that convective flows could be driven towards two distinct marginally stable states, one of which we associate with MADs. We confirm the plausibility of our theoretical model by comparing its quantitative predictions to simulations of both MAD and SANE (standard and normal evolution; strongly magnetized but not ‘arrested’) discs, and suggest a set of criteria that could help to distinguish MADs from other accretion states. Contrary to previous claims in the literature, we argue that MRI is not suppressed in MADs and is probably responsible for the existence of the strong toroidal field.