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Lorentz invariance violation (LIV) is often described by dispersion relations of the form E i2 = m i2+p i2+δi,n E 2+n with delta different based on particle type i, with energy E, momentum p and rest mass m. Kinematics and energy thresholds of interactions are modified once the LIV terms become comparable to the squared masses of the particles involved. Thus, the strongest constraints on the LIV coefficients δi,n tend to come from the highest energies. At sufficiently high energies, photons produced by cosmic ray interactions as they propagate through the Universe could be subluminal and unattenuated over cosmological distances. Cosmic ray interactions can also be modified and lead to detectable fingerprints in the energy spectrum and mass composition observed on Earth. The data collected at the Pierre Auger Observatory are therefore possibly sensitive to both the electromagnetic and hadronic sectors of LIV. In this article, we explore these two sectors by comparing the energy spectrum and the composition of cosmic rays and the upper limits on the photon flux from the Pierre Auger Observatory with simulations including LIV. Constraints on LIV parameters depend strongly on the mass composition of cosmic rays at the highest energies. For the electromagnetic sector, while no constraints can be obtained in the absence of protons beyond 1019 eV, we obtain δγ,0 > -10-21, δγ,1 > -10-40 eV-1 and δγ,2 > -10-58 eV-2 in the case of a subdominant proton component up to 1020 eV. For the hadronic sector, we study the best description of the data as a function of LIV coefficients and we derive constraints in the hadronic sector such as δhad,0 < 10-19, δhad,1 < 10-38 eV-1 and δhad,2 < 10-57 eV-2 at 5σ CL.