  We study the electronic band structure and the topological property of the
twisted double bilayer graphene, or a pair of AB-stacked bilayer graphenes
rotationally stacked on top of each other. We consider two different
arrangements, AB-AB and AB-BA, which differ in the relative orientation. For
each system, we calculate the energy band and the valley Chern number using the
continuum Hamiltonian. We show that the AB-AB and the AB-BA have similar band
structures, while the Chern numbers associated with the corresponding bands are
completely different. In the absence of the perpendicular electric field, in
particular, the AB-AB system is a trivial insulator when the Fermi energy is in
a gap, while the AB-BA is a valley Hall insulator. Also, the lowest electron
and hole bands of the AB-AB are entangled by the symmetry protected band
touching points, while they are separated in the AB-BA. In both cases, the
perpendicular electric field immediately opens an energy gap at the charge
neutral point, where the electron branch becomes much narrower than the hole
branch, due to the significant electron-hole asymmetry.
