The vertical evolution of galactic discs is governed by the sub-structures within them. Several of these features, including bulges and kinematically distinct discs, are best studied in edge-on galaxies, as the viewing angle allows the easier separation of component light. For this work, we examined the diversity of kinematic sub-structure present in the first 12 galaxies observed from the GECKOS survey, a VLT/MUSE large programme providing a systematic study of 36 edge-on Milky Way-mass disc galaxies. Employing the NGIST analysis pipeline, we derived the mean luminosity-weighted line-of-sight stellar velocity (V⋆), velocity dispersion (σ⋆), skew (h3), and kurtosis (h4) for the sample, and examined 2D maps and 1D line profiles. Common clear kinematic signatures were observed: all galaxies display h3 – V⋆ sign mismatches in the outer disc regions consistent with a (quasi-)axisymmetric, rotating disc of stars. After scrutinising visual morphologies, we found that the majority of this sample (8/12) possess boxy-peanut bulges and host the corresponding kinematic structure predicted for stellar bars viewed in projection. Inferences were made on the bar viewing angle with respect to the line of sight from the strength of these kinematic indicators; we found one galaxy whose bar is close to side-on with respect to the observer, and two that are close to end-on. Four galaxies exhibit strong evidence for the presence of nuclear discs, including central h3–V⋆ profile anti-correlations, croissant-shaped central depressions in σ⋆ maps, strong gradients in h3, and positive h4 plateaus over the expected nuclear disc extent. The strength of the h3 feature corresponds to the size of the nuclear disc, measured from the h3 turnover radius, taking into account geometric effects. We can explain the features within the kinematic maps of the four unbarred galaxies via disc structure(s) alone. We do not find any need to invoke the existence of dispersion-dominated bulges in any of the sample galaxies. Obtaining the specialised data products for this paper and the broader GECKOS survey required significant development of existing integral field spectroscopic (IFS) analysis tools. Therefore, we also present the NGIST pipeline: a modern, sophisticated, and easy-to-use pipeline for the analysis of galaxy IFS data, and the key tool employed by the GECKOS survey for producing value-added data products. We conclude that the variety of kinematic sub-structures seen in GECKOS galaxies requires a contemporary view of galaxy morphology, expanding on the traditional view of galaxy structure, and uniting the kinematic complexity observed in the Milky Way with the extragalactic.