  The recent observation of superconductivity in proximity to an insulating
phase in twisted bilayer graphene (TBG) at small `magic' twist angles has been
linked to the existence of nearly-flat bands, which make TBG a fresh playground
to investigate the interplay between correlations and superconductivity. The
low-energy narrow bands were shown to be well-described by an effective
tight-binding model on the honeycomb lattice (the dual of the triangular
Moir\'e superlattice) with a local orbital degree of freedom. In this paper, we
perform a strong-coupling analysis of the proposed $\left(p_{x},\,p_{y}\right)$
two-orbital extended Hubbard model on the honeycomb lattice. By decomposing the
interacting terms in the particle-particle and particle-hole channels, we
classify the different possible superconducting, magnetic, and charge
instabilities of the system. In the pairing case, we pay particular attention
to the two-component ($d$-wave) pairing channels, which admit vestigial phases
with nematic or chiral orders, and study their phenomenology. Furthermore, we
explore the strong-regime by obtaining a simplified spin-orbital exchange model
which may describe a putative Mott-like insulating state at quarter-filling.
Our mean-field solution reveals a rich intertwinement between ferro- and
antiferro-magnetic orders with different types of nematic and magnetic orbital
orders. Overall, our work provides a solid framework for further investigations
of the phase diagram of the two-orbital extended Hubbard model in both strong-
and weak-coupling regimes.
