The central exclusive production of charged-hadron pairs in pp collisions at a center-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 twopion system in the resonance-free region, m(pi+pi-) < 0.7 or m(pi+pi-) > 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 |y| < 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, protonpomeron and meson-pomeron form factors, pomeron trajectory and intercept, and coefficients of diffractive eigenstates of the proton.