  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.
