  The nature of the insulating and superconducting states in twisted bilayer
graphene systems is intensely debated. While many works seek for explanations
in the few flat bands near the Fermi level, theory and a number of experiments
suggest that nontwisted bilayer graphene systems do exhibit - or are at least
close to - an ordered, insulating ground state related to antiferromagnetic
ordering. Here we investigate in which ways this magnetic ordering scenario is
affected by the slight twisting between the layers. We find that at charge
neutrality the ordering tendencies of twisted systems interpolate between those
of untwisted AA and AB stacked bilayers at intermediate temperatures, while at
lower temperatures of the order of typical flat-band dispersion energies, the
ordering tendencies are even enhanced for the twisted systems. The preferred
order at charge neutrality still exhibits an antiferromagnetic spin
arrangement, with ordered moments alternating on the carbon-carbon bonds, with
an enveloping variation on the moir\'e scale. This ordering can be understood
as inherited from the untwisted systems. However, even in the RPA analysis, the
possible low-energy behaviors are quite versatile, and slight doping of one or
more electrons per moir\'e cell can take the system into a, potentially
flat-band induced, ferromagnetic phase.
