This dissertation introduces a series of new two-sample tests of distributional equality. The new tests are a generalization of the Syrjala (1996) test and make use of both rotations and toroidal shifts of the data. The new tests exhibit stability across a variety of explored test statistics. The test which employs both rotations and toroidal shifts overcomes many of the limitations of the original Syrjala test, and can be used for a variety of two-sample continuous bivariate data. Furthermore, several of the new tests are shown to be competitive or better than other modern techniques via simulation experiments. One of the new tests is applied to a new study in eye-tracking and postural stability assessment, called the Utah State University (USU) Posture Study. The setup, data collection, and data preprocessing of the USU Posture Study are also provided. These new tests, called the modified Syrjala tests, are made available via the distdiffR package for the R software environment for statistical computing and graphics. A vignette and user manual for the package are also provided.