  We formulate the chiral decomposition rules that govern the electronic
structure of a broad family of twisted $N+M$ multilayer graphene configurations
that combine arbitrary stacking order and a mutual twist. We show that at the
magic angle in the chiral limit the low-energy bands of such systems are
composed of chiral pseudospin doublets which are energetically entangled with
two flat bands per valley induced by the moir\'e superlattice potential. The
analytic analysis is supported by explicit numerical calculations based on
realistic parameterization. We further show that applying vertical displacement
fields can open up energy gaps between the pseudospin doublets and the two flat
bands, such that the flat bands may carry nonzero valley Chern numbers. These
results provide guidelines for the rational design of various topological and
correlated states in generic twisted graphene multilayers.
