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.
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