— In this article, a complementary glide symmetric spoof surface plasmon polariton leaky wave antenna (SSPP-LWA) with a high scanning rate and full-angle scanning capability is presented. The novelty of this design is attributed to the combination of the complementary glide symmetric cell SSPP structure with the odd-mode feeding, so as to achieve the purpose of increasing the scanning rate and full-angle scanning. A larger gradient of dispersion is obtained by gliding the symmetric cell structure with odd-mode excitation, and the space harmonics are excited to form radiation by periodically loading the upper and lower symmetric metal branches on both sides of the SSPP transmission line (TL). A theoretical approach on the full-angle continuous scanning is presented to analyze the conditions of single-harmonic operation, the effect of the current distribution of odd and even modes on the radiation in the endfire direction, and predict its radiation direction pattern by theoretical calculations. In order to verify the proposed method, we processed and fabricated the prototype of the proposed LWA, and the theoretical calculation, simulation, and measurement results are in good agreement. The total length of the proposed LWA is 5.36 wavelengths, and it is able to achieve a full-angle scanning from −90 to +90 over a bandwidth of 2.1 GHz (4.35–6.45 GHz), with a high scanning rate of 4.6/%. It also has the advantages of low profile, high average gain (10.8 dBi), high average radiation efficiency (90%), low sidelobe (<−10 dB), and easy fabrication.