  We examine possible ordered states of AA stacked bilayer graphene arising due
to electron-electron coupling. We show that under certain assumptions the
Hamiltonian of the system possesses an SU(4) symmetry. The multicomponent order
parameter is described by a $4\times4$ matrix $\hat{Q}$, for which a mean-field
self-consistency equation is derived. This equation allows Hermitian and
non-Hermitian solutions. Hermitian solutions can be grouped into three
topologically-distinct classes. First class corresponds to the charge density
wave. Second class includes spin density wave, valley density wave, and
spin-valley density wave. An ordered state in the third class is a combination
of all the aforementioned density-wave types. For anti-Hermitian $\hat{Q}$ the
ordered states are characterized by spontaneous inter-layer loop currents
flowing in the bilayer. Depending on the topological class of the solution
these currents can carry charge, spin, valley, and spin-valley quanta. We also
discuss the special case when matrix $\hat{Q}$ is not Hermitian and not
anti-Hermitian. Utility and weak points of the proposed SU(4)-based
classification scheme of the ordered states are analyzed.
