In van der Waals (vdW) heterostructures consisting of atomically thin crystals layered on top of one another, lattice mismatch and rotation between the layers can result in long-wavelength moire superlattices. These moire patterns can drive notable band structure reconstruction of the composite material, leading to a wide range of emergent phenomena including superconductivity(1-3), magnetism(4), fractional Chern insulating states(5) and moire excitons(6-9). Here, we investigate devices consisting of monolayer graphene encapsulated between two crystals of boron nitride (BN), in which the rotational alignment of all three components is controlled. We find that bandgaps in the graphene arising from perfect rotational alignment with both BN layers can be modified considerably depending on whether the relative orientation of the two BN layers is 0 degrees or 60 degrees, suggesting a tunable transition between the absence or presence of inversion symmetry in the heterostructure. Small deviations (<1 degrees) 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 symmetry and electronic properties of the composite heterostructure.