A celebrated theorem of Salmon and Cayley states that there are 27 lines on a smooth complex cubic surface. This result is the most well-known from a large body of 19th century work on cubic surfaces and the investigation of geometric quantities attached to them. We will provide a modern retelling of the classical story, recasting much of this in the language of W(E_6)-sets. Given a geometric symmetry of the cubic surface, for instance asking the cubic surface to be defined by a symmetric polynomial, leads to a program of “equivariant enumerative geometry” in which we count solutions to an enumerative problem together with their symmetries. In this talk we will examine symmetric cubic surfaces as a testbed for the interplay between symmetry and monodromy, enriching classical counts of lines and tritangents.