  We investigate invariant canonical transformations of a spatially covariant
scalar-tensor theory of gravity, called the XG theory, by which the action or
the Hamiltonian and the primary constraints keep their forms invariant. We
derive the Hamiltonian in a non perturbative manner and complete the
Hamiltonian analysis for all regions of the theory. We confirm that the theory
has at most 3 degrees of freedom in all regions of the theory as long as the
theory has the symmetry under the spatial diffeormorphism. Then, we derive the
invariant canonical transformation by using the infinitesimal transformation.
The invariant metric transformation of the XG theory contains a vector product
as well as the disformal transformation. The vector product and the disformal
factor can depend on the higher order derivative terms of the scalar field and
the metric. In addition, we discover the invariant canonical transformation
which transforms the momentum of the metric. Using the invariant
transformation, we study the relation between the Horndeski theory and the GLPV
theory, and find that we can not obtain the arbitrary GLPV theory from the
Horndeski theory through the invariant canonical transformation we have found.
