The efficiency of natural light-harvesting complexes relies on delocalization and directed transfer of excitation energy on spatially well-defined arrangements of molecular absorbers. Coherent excitation delocalization and long-range molecular order are also central prerequisites for engineering energy flows in bioinspired devices. Double-wall cylindrical aggregates have emerged as excellent candidates that meet these criteria. So far, the experimental signatures of exciton relaxation in these tubular supramolecules could not be linked to models encompassing their entire spatial structure. On the basis of the power of two-dimensional electronic spectroscopy, we characterize the motion of excitons in the three-fold band structure of the bitubular aggregate C8S3 through temporal, energetic, and spatial attributes. Accounting for intra- as well as interwall electronic interactions in the framework of a Frenkel exciton basis, we employ numerical computations using inhomogeneous and homogeneous microscopic models. The calculations on large but finite structures identify disorder-induced effects, which become increasingly relevant for higher energy states and give insight into the topology of the excited state manifold. Calculations in the infinite homogeneous limit capture the phenomena evidenced in the experimental two-dimensional patterns. Our results provide a basis for understanding recently reported correlated fluctuations of excitonic absorption bands and interband coherences in tubular aggregates.