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

Commensurate and incommensurate double moire interference in graphene encapsulated by hexagonal boron nitride