  A characteristic feature of the 3d plaquette Ising model is its planar
subsystem symmetry. The quantum version of this model has been shown to be
related via a duality to the X-Cube model, which has been paradigmatic in the
new and rapidly developing field of fractons. The relation between the 3d
plaquette Ising and the X-Cube model is similar to that between the 2d quantum
transverse spin Ising model and the Toric Code. Gauging the global symmetry in
the case of the 2d Ising model and considering the gauge invariant sector of
the high temperature phase leads to the Toric Code, whereas gauging the
subsystem symmetry of the 3d quantum transverse spin plaquette Ising model
leads to the X-Cube model. A non-standard dual formulation of the 3d plaquette
Ising model which utilises three flavours of spins has recently been discussed
in the context of dualising the fracton-free sector of the X-Cube model. In
this paper we investigate the classical spin version of this non-standard dual
Hamiltonian and discuss its properties in relation to the more familiar
Ashkin-Teller-like dual and further related dual formulations involving both
link and vertex spins and non-Ising spins.
