  The central exclusive production of charged hadron pairs in pp collisions at
a centre-of-mass energy of 13 TeV is examined, based on data collected in a
special high-$\beta^*$ run of the LHC. Events are selected by requiring both
scattered protons detected in the TOTEM Roman pots, exactly two oppositely
charged identified particles in the CMS silicon tracker, and the
energy-momentum balance of these four particles. The nonresonant continuum
processes are studied with the invariant mass of the centrally produced
two-pion system in the resonance-free region, $m < 0.7~\mathrm{GeV}$ or $m >
1.8~\mathrm{GeV}$. Differential cross sections as functions of the azimuthal
angle between the surviving protons, squared four-momenta, and two-hadron
invariant mass are measured in a wide region of scattered proton transverse
momenta $0.2~\mathrm{GeV} < p_\text{1,T}, p_\text{2,T} < 0.8~\mathrm{GeV}$ and
for hadron rapidities $|y| < 2$. A rich structure of interactions related to
double pomeron exchange emerges. The parabolic minimum in the distribution of
the two-proton azimuthal angle is observed for the first time. It can be
understood as an effect of additional pomeron exchanges between the protons
from the interference between the bare and the rescattered amplitudes. After
model tuning, various physical quantities related to the pomeron cross section,
proton-pomeron and hadron-pomeron form factors, trajectory slopes and
intercepts, as well as coefficients of diffractive eigenstates of the proton
are determined.
