  A peculiar manifestation of orbital angular momentum is proposed for a
zeolite-templated carbon system, C36H9. The structure, being a network of
nanoflakes in the shape of a "pinwheel", lacks inversion symmetry. While the
unit cell is large, the electronic structure obtained with a first-principles
density functional theory and captured as an effective tight-binding model in
terms of maximally-localized Wannier functions, exhibits an unusual feature
that the valence band top comes from two chiral states having orbital magnetic
momenta of $\pm 1$. The noncentrosymmetric lattice structure then makes the
band dispersion asymmetric, as reminiscent of, but totally different from,
spin-orbit systems. The unusual feature is predicted to imply a current-induced
orbital magnetism when holes are doped.
