We present kinematic orientations and high-resolution (150 pc) rotation curves for 67 main-sequence star-forming galaxies surveyed in CO (2-1) emission by PHANGS-ALMA. Our measurements are based on the application of a new fitting method tailored to CO velocity fields. Our approach identifies an optimal global orientation as a way to reduce the impact of nonaxisymmetric (bar and spiral) features and the uneven spatial sampling characteristic of CO emission in the inner regions of nearby galaxies. The method performs especially well when applied to the large number of independent lines of sight contained in the PHANGS CO velocity fields mapped at 1″ resolution. The high-resolution rotation curves fitted to these data are sensitive probes of mass distribution in the inner regions of these galaxies. We use the inner slope as well as the amplitude of our fitted rotation curves to demonstrate that CO is a reliable global dynamical mass tracer. From the consistency between photometric orientations from the literature and kinematic orientations determined with our method, we infer that the shapes of stellar disks in the mass range of log = 9.0-10.9 probed by our sample are very close to circular and have uniform thickness.