We discuss the chiral decomposition of non-symmetric stacking structures. It is shown that the low-energy electronic structure of a Bernal stacked graphene multilayer deposited on h-BN consists of chiral pseudospin doublets. N-layer graphene stacks on the h-BN layer have N/2 effective bilayer graphene systems and one effective h-BN layer if N is even or [Formula: see text] effective graphene bilayers plus one graphene monolayer modified by an h-BN layer if N is odd. We present the decomposition procedure and derive the recurrence relations for the effective parameters characterizing the chiral subsystems. In this case, the effective parameters consist of interlayer couplings and on-site potentials in contrast to pure graphene multilayer systems where only interlayer couplings are modified. We apply this procedure to discuss the Klein tunnelling phenomena and quantitatively compare the results with pure graphene multilayer systems.