In heterostructures consisting of atomically thin crystals layered on top of one another, lattice mismatch or rotation between the layers results in long-wavelength moire superlattices. These moire patterns can drive significant band structure reconstruction of the composite material, leading to a wide range of emergent phenomena including superconductivity, magnetism, fractional Chern insulating states, and moire excitons. Here, we investigate monolayer graphene encapsulated between two crystals of boron nitride (BN), where the rotational alignment between all three components can be varied. We find that band gaps in the graphene arising from perfect rotational alignment with both BN layers can be modified substantially depending on whether the relative orientation of the two BN layers is 0 or 60 degrees, suggesting a tunable transition between the absence or presence of inversion symmetry in the heterostructure. Small deviations ($<1^{circ}$) from perfect alignment of all three layers leads to coexisting long-wavelength moire potentials, resulting in a highly reconstructed graphene band structure featuring multiple secondary Dirac points. Our results demonstrate that the interplay between multiple moire patterns can be utilized to controllably modify the electronic properties of the composite heterostructure.
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