Non-identical moir\'e twins in bilayer graphene

The superlattice obtained by aligning a monolayer graphene and boron nitride (BN) inherits from the hexagonal lattice a sixty degrees periodicity with the layer alignment. It implies that, in principle, the properties of the heterostructure must be identical for 0$^{\circ}$ and 60$^{\circ}$ of layer alignment. Here, we demonstrate, using dynamically rotatable van der Waals heterostructures, that the moir\'e superlattice formed in a bilayer graphene/BN has different electronic properties at 0$^{\circ}$ and 60$^{\circ}$ of alignment. Although the existence of these non-identical moir\'e twins is explained by different relaxation of the atomic structures for each alignment, the origin of the observed valley Hall effect remains to be explained. A simple Berry curvature argument do not hold to explain the hundred and twenty degrees periodicity of this observation. Our results highlight the complexity of the interplay between mechanical and electronic properties on moir\'e structure and the importance of taking into account atomic structure relaxation to understand its electronic properties.