A measure of "clusterability" serves as the basis of a new methodology designed to preserve cluster structure in a reduced dimensional space. Similar to principal component analysis, which finds the direction of maximal variance in multivariate space, principal cluster axes find the direction of maximum clusterability in multivariate space. Furthermore, the principal clustering approach falls into the class of projection pursuit techniques. Comparisons are made with existing methodologies both in a simulation study and analysis of real-world data sets. Furthermore, a demonstration of how to interpret the results of the principal cluster axes is provided on the analysis of Supreme Court voting data and similarities between the interpretation of competing procedures (e.g., factor analysis and principal component analysis) are provided. In addition to the Supreme Court analysis, we analyze several data sets often used to test cluster analysis procedures, including Fisher's Iris data, Agresti's Crab data, and a data set on glass fragments. Finally, discussion is provided to help determine when the proposed procedure will be the most beneficial to the researcher. (Contains 5 footnotes, 5 tables, and 8 figures.)