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 fully understood. For example, where is the exciton formed within a conjugated segment, is it always situated on the same repeat units? Here, we introduce structurally-rigid molecular spoked wheels, 6 nanometres in diameter, as a model of extended pi-conjugation. Single-molecule fluorescence reveals random exciton localisation, leading to temporally-varying emission polarisation. Initially, this random localisation arises after every photon absorption event because of temperature independent spontaneous symmetry breaking. These fast fluctuations are slowed to millisecond timescales following prolonged illumination. Intramolecular heterogeneity is revealed in cryogenic spectroscopy by jumps in transition energy, however, emission polarisation can also switch without a spectral jump occurring, implying long-range homogeneity in local dielectric environment.