Chiral spin textures of a ferromagnetic layer in contact to a heavy non-magnetic metal, such as Néel-type domain walls and skyrmions, have been studied intensively because of their potential for future nanomagnetic devices. The Dyzaloshinskii-Moriya interaction (DMI) is an essential phenomenon for the formation of such chiral spin textures. In spite of recent theoretical progress aiming at understanding the microscopic origin of the DMI, an experimental investigation unravelling the physics at stake is still required. Here we experimentally demonstrate the close correlation of the DMI with the anisotropy of the orbital magnetic moment and with the magnetic dipole moment of the ferromagnetic metal in addition to Heisenberg exchange. The density functional theory and the tight-binding model calculations reveal that inversion symmetry breaking with spin-orbit coupling gives rise to the orbital-related correlation. Our study provides the experimental connection between the orbital physics and the spin-orbit-related phenomena, such as DMI.