  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. The nonresonant continuum processes are
studied with the invariant mass of the centrally produced two-pion system in
the resonance-free region, $m_{\pi^+\pi^-}$ $\lt$ 0.7 GeV or $m_{\pi^+\pi^-}$
$\gt$ 1.8 GeV. Differential cross sections as functions of the azimuthal angle
between the surviving protons, squared exchanged four-momenta, and
$m_{\pi^+\pi^-}$ are measured in a wide region of scattered proton transverse
momenta, between 0.2 and 0.8 GeV, and for pion rapidities $\lvert y\rvert$
$\lt$ 2. A rich structure of interactions related to double-pomeron exchange is
observed. A parabolic minimum in the distribution of the two-proton azimuthal
angle is observed for the first time. It can be interpreted 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 are determined that are related to the pomeron cross section,
proton-pomeron and meson-pomeron form factors, pomeron trajectory and
intercept, and coefficients of diffractive eigenstates of the proton.
