Two-way twisting of a confined monolayer: orientational ordering within the van der Waals gap between graphene and its crystalline substrate

Two-dimensional confinement of lattices produces a variety of order and disorder phenomena. When the confining walls have atomic granularity, unique structural phases are expected, of relevance in nanotribology, porous materials or intercalation compounds where \textit{e.g.} electronic states can emerge accordingly. The interlayer's own order is frustrated by the competing interactions exerted by the two confining surfaces. We revisit the concept of orientational ordering, introduced by Novaco and McTague to describe the twist of incommensurate monolayers on crystalline surfaces. We predict a two-way twist of the monolayer as its density increases. We discover such a behavior in alkali atom monolayers (sodium, cesium) confined between graphene and an iridium surface, using scanning tunneling microscopy and electron diffraction.