While the kinematics of galaxies up to z similar to 3 have been characterized in detail, only a handful of galaxies at high redshift (z > 4) have been examined in such a way. The Atacama Large Millimeter/submillimeter Array (ALMA) Large Program to INvestigate [C II] at Early times (ALPINE) survey observed a statistically significant sample of 118 star-forming main-sequence galaxies at z = 4.4-5.9 in [C. II]158 mu m emission, increasing the number of such observations by nearly 10 x . A preliminary qualitative classification of these sources revealed a diversity of kinematic types (i.e. rotators, mergers, and dispersion-dominated systems). In this work, we supplement the initial classification by applying quantitative analyses to the ALPINE data: a tilted ring model (TRM) fitting code ((3D)BAROLO), a morphological classification (Gini-M-20), and a set of disc identification criteria. Of the 75 [CM-detected ALPINE galaxies, 29 are detected at sufficient significance and spatial resolution to allow for TRM fitting and the derivation of morphological and kinematic parameters. These 29 sources constitute a high-mass subset of the ALPINE sample (M-* > 10(9.5) M-circle dot). We robustly classify 14 of these sources (six rotators, five mergers, and three dispersion-dominated systems); the remaining sources showing complex behaviour. By exploring the G-M-20 of z > 4 rest-frame far-infrared and [CII] data for the first time, we find that our 1 arcsec similar to 6 kpc resolution data alone are insufficient to separate galaxy types. We compare the rotation curves and dynamical mass profiles of the six ALPINE rotators to the two previously detected z similar to 4-6 unlensed main-sequence rotators, finding high rotational velocities (similar to 50-250 km s(-1)) and a diversity of rotation curve shapes.