The kagome lattice is a paragon of geometrical frustration, long-studied for its association with novel ground states including spin liquids. Many recently synthesized kagome materials feature rare-earth ions, which may be expected to exhibit highly anisotropic exchange interactions. The consequences of this combination of strong exchange anisotropy and extreme geometrical frustration are yet to be fully understood. Here, we establish a general picture of the interactions and resulting ground states arising from nearest-neighbor exchange anisotropy on the kagome lattice. We determine a generic anisotropic exchange Hamiltonian from symmetry arguments. In the high-symmetry case where reflection in the kagome plane is a symmetry of the system, the generic nearest-neighbor Hamiltonian can be locally defined as an XYZ model with out-of-plane Dzyaloshinskii-Moriya interactions. We proceed to study its phase diagram in the classical limit, making use of an exact reformulation of the Hamiltonian in terms of irreducible representations (irreps) of the lattice symmetry group. This reformulation in terms of irreps naturally explains the threefold mapping between three families of models supporting spin liquids, as recently studied by the present authors [Nat. Commun. 7, 10297 (2016)]. In addition, a number of unusual states are stabilized in the regions where different forms of ground-state order compete, including a stripy phase with a local Z8 symmetry and a classical analog of a chiral spin liquid. As a peculiar property of the kagome lattice, the generic model turns out to be a fruitful hunting ground for the coexistence, in the same ground-state configuration, of multiple forms of long-range magnetic orders. In exotic instances, partial long-range order may also coexist in the ground state with a finite fraction of disordered spin degrees of freedom. These results are compared and contrasted with those obtained on the pyrochlore lattice, and connection is made with recent progress in understanding quantum models with S=1/2.< Réduire