  In the study of three-dimensional gapped models, two-dimensional gapped
states should be considered as a free resource. This is the basic idea
underlying the notion of `foliated fracton order' proposed in Phys. Rev. X 8,
031051 (2018). We have found that many of the known type I fracton models,
although they appear very different, have the same foliated fracton order,
known as `X-cube' order. In this paper, we identify three-dimensional fracton
models with different kinds of foliated fracton order. Whereas the X-cube order
corresponds to the gauge theory of a simple paramagnet with subsystem planar
symmetry, the novel orders correspond to twisted versions of the gauge theory
for which the system prior to gauging has nontrivial order protected by the
planar subsystem symmetry. We present constructions of the twisted models and
demonstrate that they possess nontrivial order by studying their fractional
excitation contents.
