We study the properties of neutral hydrogen atoms precipitating onto the upper atmosphere of Mars. Energetic neutral atoms (ENAs) are produced by the charge exchange process between protons of solar wind (both upstream and shocked) as well as planetary origins and the Martian exospheric neutrals. Using a global hybrid plasma model for Mars-solar wind interaction combined with an up-to-date exosphere model of Mars, we calculate the fluxes, spatial distributions, energy spectra, and direction distributions of hydrogen ENAs (H-ENAs) at the Martian exobase for each source proton population. H-ENAs originating from the upstream solar wind region and the magnetosheath dominate the precipitation. Two percent of the solar wind flux penetrates through the magnetic barrier as H-ENAs under solar minimum conditions. The precipitating solar wind H-ENA flux is axially symmetric about Sun-Mars line, while the magnetosheath and planetary H-ENAs have higher fluxes and a more-extended precipitation area in the hemisphere where the convection electric field is pointing away from the planet, causing a significant precipitation beyond the terminator. The observed asymmetry is consistently explained by an asymmetric plasma flow in the dayside magnetosheath. The solar wind dynamic pressure increases the solar wind H-ENA precipitation normalized by the upstream proton flux, due to a closer bow shock position and thus a higher exospheric column density for charge exchange. The spatial distribution of the magnetosheath solar wind and planetary H-ENAs becomes more axially symmetric with increased dynamic pressure. The solar wind interaction with Mars exhibits more gas-dynamic-like signatures for higher dynamic pressure.