  In heterostructures consisting of atomically thin crystals layered on top of
one another, lattice mismatch or rotation between the layers results in
long-wavelength moir\'e superlattices. These moir\'e 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 moir\'e 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 moir\'e potentials, resulting
in a highly reconstructed graphene band structure featuring multiple secondary
Dirac points. Our results demonstrate that the interplay between multiple
moir\'e patterns can be utilized to controllably modify the electronic
properties of the composite heterostructure.
