  The kagome lattice is a paragon of geometrical frustration, long-studied for
its association with novel ground-states including spin liquids (SLs). 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 (GSs) arising from
nearest neighbour 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-neighbour Hamiltonian can be locally defined as a
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 three-fold mapping between SLs recently studied on
kagome by the present authors [Nat Commun 7, 10297 (2016)]. In addition, a
number of unusual states are stabilised in the regions where different forms of
GS order compete, including a stripy phase with a local Z8 symmetry and a
classical analogue of a chiral SL. This generic Hamiltonian also turns out to
be a fruitful hunting ground for coexistence of different forms of magnetic
order, or of order and disorder, which we find is a particular property of the
kagome lattice arising from the odd number of spins per frustrated unit. 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
