Orientationally ordered structures of two-dimensional (2D) C-60 at low temperature have been investigated theoretically and experimentally. Using total energy optimization with a phenomenological potential, we find the ground state is a close packed hexagonal lattice in which all the molecules have the same orientation. Several local minima of the potential energy surface are found to be associated with other 1 x 1 lattices as well as 2 x 2 lattices. The energies of the orientational domain boundaries of the 1 x 1 lattices are also computed, and two kinds of which yield negative values. A majority of these theoretical findings are confirmed by our low-temperature scanning tunneling microscopy study of a 2D C-60 array supported on a self-assembled monolayer.