Conjugated polymers offer potential for many diverse applications, but we still lack a fundamental microscopic understanding of their electronic structure. Elementary photoexcitations (excitons) span only a few nanometres of a molecule, which itself can extend over microns, and how their behaviour is affected by molecular dimensions is not immediately obvious. For example, where is the exciton formed within a conjugated segment and is it always situated on the same repeat units? Here, we introduce structurally rigid molecular spoked wheels, 6 nm in diameter, as a model of extended π conjugation. Single-molecule fluorescence reveals random exciton localization, which leads to temporally varying emission polarization. Initially, this random localization arises after every photon absorption event because of temperature-independent spontaneous symmetry breaking. These fast fluctuations are slowed to millisecond timescales after prolonged illumination. Intramolecular heterogeneity is revealed in cryogenic spectroscopy by jumps in transition energy, but emission polarization can also switch without a spectral jump occurring, which implies long-range homogeneity in the local dielectric environment.