The sensitivity of the OD stretching band as a probe to detect HDO in astrophysical ice is discussed based on IR laboratory spectra of HDO molecules embedded in H2O ice. This band is extremely broad and tends to disappear into the absorption continuum of H2O for low-temperature amorphous samples. Detectable HDO/H2O ratios with this technique may range from a few percent for amorphous samples to a few per thousand in crystalline ice. These relatively high upper limits and the appreciable dependence of the band shape on temperature, which would complicate the interpretation of data from many lines of sight, decisively limit the usefulness of the technique for HDO detection in astronomical observations. The process of isotopic H/D exchange in mixed ice of H2O/D2O is also studied through the evolution of the OD band in IR spectra. Isotopic exchange starts at ~120 K and is greatly accelerated at 150 K, as crystallization proceeds in the ice. Annealed amorphous samples prove to be more favorable for isotope exchange than samples directly formed in crystalline phase. The annealing process seems to favor a polycrystalline ice morphology with a higher defect activity. These morphology differences can be of relevance for deuterium fractionation in astronomical environments.