Recent results on the diffractive production of vector mesons in photon-proton reactions at HERA are challenging contemporary understanding of diffractive processes and of hadron structure. Following a brief overview of selected results obtained from the measurement programs of the H1 and ZEUS collaborations during the first eight years of operation, we concentrate on the experimental and phenomenological particulars relating to a recent observation of power-law scaling with momentum transfer in semi-exclusive vector-meson photoproduction. The combination of the observed power and the polarization of the vector meson appear to violate the helicity selection rules of perturbative QCD. This observation fits into a pattern of HERA results pointing to contributions from a point-like transverse-to-transverse vacuum-exchange transition which is difficult to reconcile with QCD.