We show how to extract the orbital contribution to the transport eigenchannels from a first-principles quantum transport calculation in a nanoscopic conductor. This is achieved by calculating and diagonalizing the first-principles transmission matrix reduced to selected scattering cross-sections. As an example, the orbital nature of the eigenchannels in the case of Ni nanocontacts is explored, stressing the difficulties inherent to the use of non-orthogonal basis sets and first-principles Hamiltonians. ; Comment: 5 pages, 5 figurs; replaced with final version, introduction revised; to be published in PRB