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The engineering of the energy dispersion of polaritons in microcavities through nanofabrication or through the exploitation of intrinsic material and cavity anisotropies has demonstrated many intriguing effects related to topology and emergent gauge fields such as the anomalous quantum Hall and Rashba effects. Here we show how we can obtain different Berry curvature distributions of polariton bands in a strongly coupled organic–inorganic two-dimensional perovskite single-crystal microcavity. The spatial anisotropy of the perovskite crystal combined with photonic spin–orbit coupling produce two Hamilton diabolical points in the dispersion. An external magnetic field breaks time-reversal symmetry owing to the exciton Zeeman splitting and lifts the degeneracy of the diabolical points. As a result, the bands possess non-zero integral Berry curvatures, which we directly measure by state tomography. In addition to the determination of the different Berry curvatures of the multimode microcavity dispersions, we can also modify the Berry curvature distribution, the so-called band geometry, within each band by tuning external parameters, such as temperature, magnetic field and sample thickness. Engineering the energy dispersion of polaritons in microcavities can yield intriguing effects such as the anomalous quantum Hall and Rashba effects. Now, different Berry curvature distributions of polariton bands are obtained in a strongly coupled organic–inorganic two-dimensional perovskite single-crystal microcavity and can be modified via temperature and magnetic field variation.