We discuss the possibility to use the pp → ppфф process in identifying the odderon exchange. So far, there is no unambiguous experimental evidence for the odderon, the charge conjugation C = −1 counterpart of the C = +1 pomeron. Last year, the results of the TOTEM collaboration suggest that the odderon exchange can be responsible for a disagreement of theoretical calculations and the TOTEM data for the elastic proton-proton scattering. Here, we present recent studies for the central exclusive production (CEP) of фф pairs in proton-proton collisions. We consider the pomeron-pomeron fusion to фф (PP → фф) through the continuum processes, due to the ^t- and ^u-channel reggeized ф-meson, photon, and odderon exchanges, as well as through the s-channel resonance process (PP → f2(2340) → фф). This f2 state is a candidate for a tensor glueball. The amplitudes for the processes are formulated within the tensor-pomeron and vector-odderon approach. Some model parameters are determined from the comparison to the WA102 experimental data. The odderon exchange is not excluded by the WA102 data for high фф invariant masses. The measurement of large Mфф or Ydiff events at the LHC would therefore suggest the presence of the odderon exchange. The process is advantageous, as here the odderon does not couple to protons.