A new and detailed measurement of the cross section for hard exclusive neutral-pion muopro-duction on the proton was performed in a wide kinematic region, with the photon virtuality Q2 ranging from 1 to 8 (GeV/c)2 and the Bjorken variable xBj ranging from 0.02 to 0.45. The data were collected at COMPASS at CERN using 160 GeV/c longitudinally polarized μ± beams scattering off a 2.5~m long liquid hydrogen target. From the average of the measured μ+ and μ− cross sections, the virtual photon-proton cross section is determined as a function of the squared four-momentum transfer between the initial and final state proton in the range 0.08 (GeV/c)2 < |t| < 0.64 (GeV/c)2. From its angular distribution, the combined contribution of transversely and longitudinally polarized photons are determined, as well as transverse-transverse and longitudinal-transverse interference contributions. They are studied as functions of four-momentum transfer |t|, photon virtuality Q2 and virtual-photon energy ν. The longitudinal-transverse interference contribution is found to be compatible with zero. The significant transverse-transverse interference contribution reveals the existence of a dominant contribution by transversely polarized photons. This provides clear experimental evidence for the chiral-odd GPD ET. In addition, the existence of a non-negligible contribution of longitudinally polarized photons is suggested by the |t|-dependence of the cross section at xBj < 0.1. Altogether, these results provide valuable input for future modelling of GPDs and thus of cross sections for exclusive pseudo-scalar meson production. Furthermore, they can be expected to facilitate the study of next-to-leading order corrections and higher-twist contributions.