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, $\phi$ < 0.7 GeV or $\phi$ > 1.8
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 GeV <
$p_{1,T} , p_{2,T}$ < 0.8 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.