  We calculate high-energy photoproduction of the tensor meson $f_2(1270)$ by
odderon and photon exchange in the reaction $\gamma + {\rm{p}} \to f_2(1270) +
{\rm{X}}$, where X is either the nucleon or the sum of the N(1520) and N(1535)
baryon resonances. Odderon exchange dominates except at very small transverse
momentum, and we find a cross section of about 20 nb at a centre-of-mass energy
of 20 GeV. This result is compared with what is currently known experimentally
about $f_2$ photoproduction. We conclude that odderon exchange is not ruled out
by present data. On the contrary, an odderon-induced cross section of the above
magnitude may help to explain a puzzling result observed by the E687
experiment.
