The central exclusive production of charged-hadron pairs in Formula Presented collisions at a center-of-mass energy of 13 TeV is examined, based on data collected in a special high-Formula Presented 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, Formula Presented or Formula Presented. Differential cross sections as functions of the azimuthal angle between the surviving protons, squared exchanged four-momenta, and Formula Presented are measured in a wide region of scattered proton transverse momenta, between 0.2 and 0.8 GeV, and for pion rapidities Formula Presented. 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.