Using the Ultraviolet Visual Echelle Spectrograph (UVES) at the Very Large Telescope (VLT), we study the photoionized Herbig-Haro object HH529 in two associated bowshocks: HH529II and HH529III and its nebular background emission. We spatially isolate each bowshock and separate its blueshifted emission from the nebular background thanks to the very high spectral resolution (6.5 km/s) of our data. We measure hundreds of emission lines both from collisional excitation (CELs) and recombination (RLs) in each velocity component in our wide spectral range (3100-10400 Å). In each case, we derive physical conditions (electron density and temperature) using multiple methods and chemical abundances both based on CELs and RLs in the cases of He+, O+, O++, C++ and Ne++. Dust destruction processes and the reincorporation of heavy elements to the gaseous state are detected in the shock heads of HH529. We also study the velocity structure and thermal width of the detected lines in the spectra for the different velocity components, including the nebular one. Comparatively, there are noticeable differences between the physical conditions and chemical composition of the nebular gas and the shock components. These objects are excellent laboratories to analyze the well-known Abundance Discrepancy problem (AD) and its possible origin.