Perpendicular magnetization is essential for high-density memory application using magnetic materials. High-spin polarization of conduction electrons is also required for realizing large electric signals from spin-dependent transport phenomena. The Heusler alloy is a well-known material class showing the half-metallic electronic structure. However, its cubic lattice nature favors in-plane magnetization and thus minimizes the perpendicular magnetic anisotropy (PMA), in general. This study focuses on an inverse-type Heusler alloy, Mn2-delta CoGa1+delta (MCG), with a small off-stoichiometry (delta), which is expected to be a half-metallic material. We observed a relatively large uniaxial magnetocrystalline anisotropy constant (K-u) of the order of 10(5) J/m(3) at room temperature in MCG films with a small tetragonal distortion of a few percent. A positive correlation was confirmed between the c/a ratio of lattice constants and K-u. Imaging of magnetic domains using Kerr microscopy clearly demonstrated a change in the domain patterns along with K-u. X-ray magnetic circular dichroism (XMCD) was employed using a synchrotron radiation soft x-ray beam to get insight into the origin of PMA. Negligible angular variation of orbital magnetic moment (Delta m(orb)) evaluated using the XMCD spectra suggested a minor role of the so-called Bruno's term to K-u. Our first-principles calculation reasonably explained the small Delta m(orb) and the positive correlation between the c/a ratio and K-u. The origin of the magnetocrystalline anisotropy was discussed based on the second-order perturbation theory in terms of the spin-orbit coupling, claiming that the mixing of the occupied up arrow- and the unoccupied down arrow-spin states is responsible for the PMA of the MCG films.