This two-part paper extends the recent results in plane elasticity to generalized plane deformation of electromagnetic thermoelastic materials. These include the correspondences between various physical contexts, invariance of stresses under a change in elastic compliance, and the reduced dependence of effective elastic compliance upon the material constants. The first part is concerned with the issues of correspondence and invariance shifts. Further results on invariance shifts and reduced independence of effective moduli are presented in Part II. We considered a cylindrical body in which the cross-sections and material moduli do not vary with the axial direction. The loading and geometric configurations are arranged so that the stress, electric displacement and magnetic induction fields are independent of the axial direction. A general solution framework is outlined. With the help of some tensor notations together with proper arrangements of the constitutive equations, the field equations of deformation of this kind are written in a compact form. The relationships between various field variables are then easily identified. This permits us to reconstruct the correspondence relations between various systems in a broader sense under all kinds of crystallographic and non-crystallographic point group symmetry. Complete forms of invariance shifts of moduli together with a proof of the completeness are given. This provides some new insights of the original discovery by Cherkaev, Lurie and Milton.