  Lorentz invariance violation (LIV) is often described by dispersion relations
of the form $E_i^2=m_i^2+p_i^2+\delta_{i,n} E^{2+n}$ 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 $\delta_{i,n}$ 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 $10^{19}$ eV, we obtain $\delta_{\gamma,0} > -10^{-21}$,
$\delta_{\gamma,1} > -10^{-40}$ eV$^{-1}$ and $\delta_{\gamma,2} > -10^{-58}$
eV$^{-2}$ in the case of a subdominant proton component up to $10^{20}$ 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
$\delta_{\mathrm{had},0} < 10^{-19}$, $\delta_{\mathrm{had},1} < 10^{-38}$
eV$^{-1}$ and $\delta_{\mathrm{had},2}< 10^{-57}$ eV$^{-2}$ at 5$\sigma$ CL.
