  The electronic and structural properties of atomically thin materials can be
controllably tuned by assembling them with an interlayer twist. During this
process, constituent layers spontaneously rearrange themselves in search of a
lowest energy configuration. Such relaxation phenomena can lead to unexpected
and novel material properties. Here, we study twisted double trilayer graphene
(TDTG) using nano-optical and tunneling spectroscopy tools. We reveal a
surprising optical and electronic contrast, as well as a stacking energy
imbalance emerging between the moir\'e domains. We attribute this contrast to
an unconventional form of lattice relaxation in which an entire graphene layer
spontaneously shifts position during fabrication. We analyze the energetics of
this transition and demonstrate that it is the result of a non-local relaxation
process, in which an energy gain in one domain of the moire lattice is paid for
by a relaxation that occurs in the other.
