  We investigate twisted double bilayer graphene (TDBG), a four-layer system
composed of two AB-stacked graphene bilayers rotated with respect to each other
by a small angle. Our ab initio band structure calculations reveal a
considerable energy gap at the charge point neutrality that we assign to the
intrinsic symmetric polarization (ISP). We then introduce the ISP effect into
the tight-binding parameterization and perform calculations on TDBG models that
include lattice relaxation effects down to very small twist angles. We identify
a narrow region around the magic angle $\theta^\circ = 1.3^{\circ}$
characterized by a manifold of remarkably flat bands gapped out from other
states even without external electric fields. To understand the fundamental
origin of the magic angle in TDBG, we construct a continuum model that points
to a hidden mathematical link to the twisted bilayer graphene (TBG) model, thus
indicating that the band flattening is a fundamental feature of TDBG, and is
not a result of external fields.
