  We discuss the possibility to use the $pp \to pp \phi \phi$ 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 results of the TOTEM collaboration suggest that the
odderon exchange can be responsible for a disagreement of theoretical
calculations and the TOTEM data for elastic proton-proton scattering. Here we
present recent studies for central exclusive production (CEP) of $\phi \phi$
pairs in proton-proton collisions. We consider the pomeron-pomeron fusion to
$\phi \phi$ ($\mathbb{P}\mathbb{P} \to \phi \phi$) through the continuum
processes, due to the $\hat{t}$- and $\hat{u}$-channel reggeized $\phi$-meson,
photon, and odderon exchanges, as well as through the $s$-channel resonance
process ($\mathbb{P}\mathbb{P} \to f_{2}(2340) \to \phi \phi$). This $f_{2}$
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 $\phi
\phi$ invariant masses. The measurement of large $M_{\phi \phi}$ or
$\rm{Y_{diff}}$ events at the LHC would therefore suggest presence of the
odderon exchange.
