We exhibit symmetries on pitch classes that are intimately related to major/minor tonality, and then recover essential aspects of such tonality from these symmetries. This fact entails a relation to Schönberg's dodecaphonic series, where their two hexachords consisting of the first and second halves exhibit such symmetries. They are by no means isolated objects; it is shown that they can be thought of as simultaneous expressions of tonal and serial instances. These symmetries define a bridge between two apparently disconnected, and even antagonistic approaches to fundamental compositional designs. We discuss applications of this approach to Schönberg's Piano Concerto, op. 42.