The nature of the insulating-like states in twisted bilayer graphene systems is an intensely debated topic currently. While many works seek for explanations solely based on the few flat bands near the Fermi level, it could be important to keep in mind that, according to theory and a number of experiments, nontwisted bilayer graphene systems do exhibit or are at least close to an ordered, insulating ground state. Here we investigate in which ways this ordering scenario is affected by the slight twisting. We find that at charge neutrality the ordering tendencies of twisted systems interpolate between those of untwisted AA and AB stacked bilayers, i.e. the instability in the twisted case is inherited from the untwisted system. The preferred order exhibits an antiferromagnetic spin arrangement, with ordered moments alternating on the length scale of the carbon-carbon bonds, with a enveloping variation on the moiré scale. However, even in the RPA analysis, the possible low-energy behaviors are quite versatile, and slight doping of two electrons per moiré cell can take the system into different phases, depending on the parameters.