We use observations from the GEMINI-N/GMOS integral-field spectrograph (IFS) to obtain spatially resolved stellar kinematics of the central $sim 1$ kpc of 20 early-type galaxies (ETGs) with stellar masses greater than $10^{11.7} M_odot$ in the MASSIVE survey. Together with observations from the wide-field Mitchell IFS at McDonald Observatory in our earlier work, we obtain unprecedentedly detailed kinematic maps of local massive ETGs, covering a scale of $sim 0.1-30$ kpc. The high ($sim 120$) signal-to-noise of the GMOS spectra enable us to obtain two-dimensional maps of the line-of-sight velocity, velocity dispersion $sigma$, as well as the skewness $h_3$ and kurtosis $h_4$ of the stellar velocity distributions. All but one galaxy in the sample have $sigma(R)$ profiles that increase towards the center, whereas the slope of $sigma(R)$ at one effective radius ($R_e$) can be of either sign. The $h_4$ is generally positive, with 14 of the 20 galaxies having positive $h_4$ within the GMOS aperture and 18 having positive $h_4$ within $1 R_e$. The positive $h_4$ and rising $sigma(R)$ towards small radii are indicative of a central black hole and velocity anisotropy. We demonstrate the constraining power of the data on the mass distributions in ETGs by applying Jeans anisotropic modeling (JAM) to NGC~1453, the most regular fast rotator in the sample. Despite the limitations of JAM, we obtain a clear $chi^2$ minimum in black hole mass, stellar mass-to-light ratio, velocity anisotropy parameters, and the circular velocity of the dark matter halo.
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