  Interference of double moire patterns of graphene (G) encapsulated by
hexagonal boron nitride (BN) can alter the electronic structure features near
the primary/secondary Dirac points and the electron-hole symmetry introduced by
a single G/BN moire pattern depending on the relative stacking arrangements of
the top/bottom BN layers. We show that strong interference effects are found in
nearly aligned BN/G/BN and BN/G/NB and obtain the evolution of the associated
density of states as a function of moire superlattice twist angles. For equal
moire periods and commensurate patterns with $\Delta \phi = 0^{\circ}$ modulo
$60^{\circ}$ angle differences the patterns can add up constructively leading
to large pseudogaps of about $\sim 0.5$ eV on the hole side or cancel out
destructively depending on their relative sliding, e.g. partially recovering
electron-hole symmetry. The electronic structure of moire quasicrystals for
$\Delta \phi =30^{\circ}$ differences reveal double moire features in the
density of states with almost isolated van Hove singularities where we can
expect strong correlations.
