  Frustrated lattices provide a versatile ground for fascinating quantum
many-body physics. Here, we study an ultracold atomic system of two-component
(pseudospin-$1/2$) Bose gases in the $p$-orbital band of a hexagonal lattice,
which exhibits frustration in an orbital setting. The potential spontaneous
symmetry-breaking orders in this complex system are classified by symmetry
analysis. We find the interplay of orbital frustration and strong interaction
leads to unconventional Mott and superfluid phases with smectic spin-orbital
intertwined orders, in spite of the complete absence of spin-orbital
interaction in the Hamiltonian. The spontaneously formed orbital polarization
forms circulating textures in the lattice, with opposite circulation in the
two-pseudospin components. Our study implies many-body correlations in a
multi-orbital setting could induce rich spin-orbital intertwined physics
without requirement of engineering spin-orbit interaction.
