  We explore the possibility of observing odderon exchange in the $pp \to pp
K^{+}K^{-}$ and $pp \to pp \mu^{+}\mu^{-}$ reactions at the LHC. We consider
the central exclusive production (CEP) of the $\phi(1020)$ resonance decaying
into $K^{+} K^{-}$ and $\mu^{+}\mu^{-}$. We compare the purely diffractive
contribution (odderon-pomeron fusion) to the photoproduction contribution
(photon-pomeron fusion). The theoretical results are calculated within the
tensor-pomeron and vector-odderon model for soft reactions. We include
absorptive corrections at the amplitude level. In order to fix the coupling
constants for the photon-pomeron fusion contribution we discuss the reactions
$\gamma p \to \omega p$ and $\gamma p \to \phi p$ including $\phi$-$\omega$
mixing. We compare our results for these reactions with the available data,
especially those from HERA. Our coupling constants for the
pomeron-odderon-$\phi$ vertex are taken from an analysis of the WA102 data for
the $p p \to p p \phi$ reaction. We show that the odderon-exchange contribution
significantly improves the description of the $pp$ azimuthal correlations and
the $dP_{t}$ "glueball-filter variable" dependence of $\phi$ CEP measured by
WA102. To describe the low-energy data more accurately we consider also
subleading processes with reggeized vector-meson exchanges. However, they do
not play a significant role at the LHC. We present predictions for two possible
types of measurements: at midrapidity and with forward measurement of protons
(relevant for ATLAS-ALFA or CMS-TOTEM), and at forward rapidities and without
measurement of protons (relevant for LHCb). We discuss the influence of
experimental cuts on the integrated cross sections and on various differential
distributions. With the corresponding LHC data one should be able to get a
decisive answer concerning the presence of an odderon-pomeron fusion
contribution in single $\phi$ CEP.
