We present an analysis of the sky distribution of neutral hydrogen (H I) deficiency and spectral asymmetry for galaxies detected by the H I Parkes All-Sky Survey (HIPASS) as a function of projected environment density. Previous studies of galaxy H I deficiency using HIPASS were sensitive to galaxies that are extremely H I rich or poor. We use an updated binning statistic for measuring the global sky distribution of H I deficiency that is sensitive to the average deficiencies. Our analysis confirms the result from previous studies that galaxies residing in denser environments, such as Virgo, are on average more H I deficient than galaxies at lower densities. However, many other individual groups and clusters are not found to be on average significantly H I poor, in contradiction to previous work. In terms of H I spectral asymmetries, we do not recover any significant trend of increasing asymmetry with environment density as found for other galaxy samples. We also investigate the correlation between H I asymmetry and deficiency, but find no variation in the mean asymmetry of galaxies that are H I rich, normal, or poor. This indicates that there is either no dependence of asymmetry on H I deficiency, or a galaxy's H I deficiency only has a small influence on the measured H I asymmetry that we are unable to observe using only integrated spectra.