Pitch and pitch-class inversion is a core concept in the pedagogy of post-tonal music. There have been many theoretical and analytical studies on this topic, beginning half a century ago with the writings of Milton Babbitt, George Perle, and David Lewin. These publications inspired many subsequent articles and model analyses of octatonic, atonal, and twelve-tone compositions. This essay outlines a self-contained module on axial symmetry that is intended to culminate with an analysis of Nuit, a solo piano composition by Jean Papineau-Couture. The module, which could be part of second-year undergraduate course, is designed to teach students to hear, recognize, and model the properties of symmetrical designs. Several factors make Nuit a strong teaching piece, among them intriguing timbres, with striking and strumming on the strings, and an inventive and sophisticated set of inversional techniques. In addition, it is an extended work that lasts approximately 16 minutes, and thus rewards a variety of approaches. Below, we review four basic types of pitch and pitch-class symmetries, and briefly define even and odd index numbers. We then illustrate the common axial techniques from the literature, and show how these techniques inform various organizational levels of the work.