  We present a detailed study of the Bose-Hubbard model in a $p$-band
triangular lattice by focusing on the evolution of orbital order across the
superfluid-Mott insulator transition. Two distinct phases are found in the
superfluid regime. One of these phases adiabatically connects the weak
interacting limit. This phase is characterized by the intertwining of axial
$p_\pm=p_x \pm ip_y$ and in-plane $p_\theta=\cos\theta p_x+\sin\theta p_y$
orbital orders, which break the time-reversal symmetry and lattice symmetries
simultaneously. In addition, the calculated Bogoliubov excitation spectrum gaps
the original Dirac points in the single-particle spectrum but exhibits emergent
Dirac points. The other superfluid phase in close proximity to the Mott
insulator with unit boson filling shows a detwined in-plane ferro-orbital
order. Finally, an orbital exchange model is constructed for the Mott insulator
phase. Its classical ground state has an emergent SO$(2)$ rotational symmetry
in the in-plane orbital space and therefore enjoys an infinite degeneracy,
which is ultimately lifted by the orbital fluctuation via the order by disorder
mechanism. Our systematic analysis suggests that the in-plane ferro-orbital
order in the Mott insulator phase agrees with and likely evolves from the
latter superfluid phase.
