Across many species where inversions have been implicated in local adaptation, genomes often evolve to contain multiple, large inversions that arise early in divergence. Why this occurs has yet to be resolved. To address this gap, we built forward-time simulations in which inversions have flexible characteristics and can invade a metapopulation undergoing spatially divergent selection for a highly polygenic trait. In our simulations, inversions typically arose early in divergence, captured standing genetic variation upon mutation, and then accumulated many small-effect loci over time. Under special conditions, inversions could also arise late in adaptation and capture locally adapted alleles. Polygenic inversions behaved similarly to a single supergene of large effect and were detectable by genome scans. Our results show that characteristics of adaptive inversions found in empirical studies (e.g. multiple large, old inversions that are outliers, sometimes overlapping with other inversions) are consistent with a highly polygenic architecture, and inversions do not need to contain any large-effect genes to play an important role in local adaptation. By combining a population and quantitative genetic framework, our results give a deeper understanding of the specific conditions needed for inversions to be involved in adaptation when the genetic architecture is polygenic. This article is part of the theme issue 'Genomic architecture of supergenes: causes and evolutionary consequences'.