Successful synthetic aperture sonar target classification depends on the "shape" of the scatterers within a target signature. This article presents a workflow that computes a target-to-target distance from persistence diagrams, since the "shape" of a signature informs its persistence diagram in a structure-preserving way. The target-to-target distances derived from persistence diagrams compare favorably against those derived from spectral features and have the advantage of being substantially more compact. While spectral features produce clusters associated to each target type that are reasonably dense and well formed, the clusters are not well-separated from one another. In rather dramatic contrast, a distance derived from persistence diagrams results in highly separated clusters at the expense of some misclassification of outliers.