  We explore the possibility of observing Odderon exchange in proton--ion
collisions at the LHC, via the ultraperipheral photoproduction of C-even
mesons. As well as the signal, we consider in detail the principle backgrounds,
due to QCD--initiated production (i.e. double Pomeron exchange) and
$\gamma\gamma$ fusion. We find that while the photon--initiated background is
dominant at very small momentum transfer, this can be effectively removed by
placing a reasonable cut on the transverse momentum of the produced meson. In
the case of QCD--initiated production, we show this is in general strongly
suppressed by the small probability of no additional particle production in the
central detector, namely the survival factor. In some scenarios, this
suppression is sufficient to permit the observation of Odderon exchange in
Pb$-p$ collisions in a clean environment, or else to place bounds on this. We
in addition identify the cases of $\pi^0$ and $\eta(548)$ production as
particularly promising channels. Here, the QCD--initiated background is absent
for $\pi^0$ due to isospin conservation and very small for $\eta(548)$ due to
its dominantly flavour octet nature and odd parity.
