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Graphene films were grown by chemical vapor deposition on Cu foil. The obtained samples were characterized by Raman spectroscopy, ellipsometry, X-ray photoelectron spectroscopy and electron back-scatter diffraction. We discuss the time-dependent changes in the samples, estimate the thickness of emerging Cu₂O beneath the graphene and check the orientation-dependent affinity to oxidation of distinct Cu grains, which also governs the manner in which the initial strong Cu-graphene coupling and strain in the graphene lattice is released. Effects of electropolishing on the quality and the Raman response of the grown graphene layers are studied by microtexture polarization analysis. The obtained data are compared with the Raman signal of graphene after transfer on glass substrate revealing the complex interaction of graphene with the Cu substrate.