  We present the analysis of physical conditions, chemical composition and
kinematic properties of two bow shocks -HH529 II and HH529 III- of the fully
photoionized Herbig-Haro object HH 529 in the Orion Nebula. The data were
obtained with the Ultraviolet and Visual Echelle Spectrograph at the 8.2m Very
Large Telescope and 20 years of Hubble Space Telescope imaging. We separate the
emission of the high-velocity components of HH529 II and III from the nebular
one, determining $n_{\rm e}$ and $T_{\rm e}$ in all components through multiple
diagnostics, including some based on recombination lines (RLs). We derive ionic
abundances of several ions, based on collisionally excited lines (CELs) and
RLs. We find a good agreement between the predictions of the temperature
fluctuation paradigm ($t^2$) and the abundance discrepancy factor (ADF) in the
main emission of the Orion Nebula. However, $t^2$ can not account for the
higher ADF found in HH 529 II and III. We estimate a 6% of Fe in the gas-phase
of the Orion Nebula, while this value increases to 14% in HH 529 II and between
10% and 25% in HH 529 III. We find that such increase is probably due to the
destruction of dust grains in the bow shocks. We find an overabundance of C, O,
Ne, S, Cl and Ar of about 0.1 dex in HH 529 II-III that might be related to the
inclusion of H-deficient material from the source of the HH 529 flow. We
determine the proper motions of HH 529 finding multiple discrete features. We
estimate a flow angle with respect to the sky plane of $58\pm 4^{\circ}$ for HH
529.
