  We consider theoretically the influence of crystalline fields on the
electronic structure of graphene placed on a layered material with reduced
symmetry and large spin-orbit coupling (SOC). We use a perturbative procedure
combined with the Slater-Koster method to derive the low-energy effective
Hamiltonian around the $K$ points and estimate the magnitude of the effective
couplings. Two simple models for the envisaged graphene-substrate hybrid
bilayer are considered, in which the relevant atomic orbitals hybridize with
either top or hollow sites of the graphene honeycomb lattice. In both cases,
the interlayer coupling to a crystal-field-split substrate is found to generate
highly anisotropic proximity spin-orbit interactions, including in-plane
'spin-valley' coupling. Interestingly, when an anisotropic intrinsic-type SOC
becomes sizeable, the bilayer system is effectively a quantum spin Hall
insulator characterized by in-plane helical edge states robust against
Bychkov-Rashba effect. Finally, we discuss the type of substrate required to
achieve anisotropic proximity-induced SOC and suggest possible candidates to
further explore crystal field effects in graphene-based heterostructures.
