A growing body of observational evidence suggests that the luminosity, photometric shape, and amount of rotational or anisotropic support in elliptical galaxies may provide vital clues to how they formed. Elliptical galaxies appear to fall into two distinct categories based on these parameters: bright, boxy-shaped, and having little or no rotation, and less luminous, disky-shaped, and having significant rotation. One viable formation scenario is the ''merger hypothesis,'' in which two disk galaxies merge to form a new elliptical galaxy. A comparison of the luminosity, photometric shape, and amount of rotation in advanced merger remnants may shed more light on the possible formation scenarios of elliptical galaxies. Yet little observational data exist for such merger remnants. This paper is the third in a series investigating the photometric and kinematic properties of a sample of 51 optically selected advanced merger remnants. Presented here are K-band isophotal shapes and spatially resolved kinematics for a subsample of 37 merger remnants. The results show that $11% of the sample is boxy and anisotropically supported, while $47% is disky and rotationally supported. The remainder of the sample shows variations among expected correlations between shape and rotation. This may suggest that the isophotal shapes are still ''in flux.'' There does appear to be a lower limit to the amount of anisotropy observed in the merger remnants. This may provide an observational diagnostic for discriminating among formation scenarios in elliptical galaxies.