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Monolayer films on graphite, remarkably diverse examples of two-dimensional matter, are now well understood in terms of semiempirical interactions. We explore the phase behavior of helium films on two variants of graphene: graphane (graphene coated with H, denoted GH) and graphene fluoride (GF). The behaviors predicted with quantum Monte Carlo differ qualitatively from those on graphite because of the different surface composition, symmetry, and spacing of the adsorption sites. On both substrates we find that the analog of the standard [radical]3 x [radical]3 R30[degrees] commensurate state on graphite is unstable. Results include a superfluid ground state for super(4)He and a fluid ground state for super(3)He, neither of which has been found for the monolayer film on any substrate; these two-dimensional fluids are anisotropic because of the symmetry imposed by the honeycomb lattice of adsorption sites. In the case of super(4)He on GF the anisotropy is as large as if the superfluid were restricted to move in a multiconnected space, along the bonds of a honeycomb lattice. The superfluid transition temperature at the groundstate density of super(4)He on GF (GH) is of order 0.25 (1.1) K. At higher coverages both an incommensurate triangular solid and a commensurate state at filling factor 2/7 are found.