Continuous transverse stub (CTS) arrays represent a high-performance and affordable solution for fixed-beam and scanning flat antenna systems, though intrinsically linearlypolarized. In these systems, the polarization is controlled using add-on polarization converters or combining cross-connected CTS arrays. This two-part paper presents a novel parallelfed CTS antenna which achieves dual-linear polarization using, instead, a single array and two orthogonally polarized excitation fields. In this first Part, it is shown that, due to modal coupling, standard parallel-plate waveguides (PPWs) are not suitable as building blocks of the proposed antenna. Therefore, an alternative waveguide, the corrugated PPW (CPPW) is introduced. The dispersion characteristics of such CPPW are investigated. Guidelines to independently design the dispersion properties of the two propagating modes and avoid the excitation of higher order modes are provided. Based on this study, the design and simulated performance of a Ka-band array of eight long slots parallel-fed by a dual-mode CPPW corporate-feed network are described. An extensive experimental validation of the concept is presented in Part II.