We apply a novel analysis of the field and angle dependence of the quantum-oscillatory amplitudes in the unconventional superconductor Sr2RuO4 to map its Fermi surface (FS) in unprecedented detail and to obtain previously inaccessible information on the band dispersion. The three quasi-2D FS sheets not only exhibit very diverse magnitudes of warping, but also entirely different dominant warping symmetries. We use the data to reassess recent results on c-axis transport phenomena.