There is a pairing on cusps of reduced Hurwitz spaces for <i>r</i> branch point covers. It pairs the reduced Nielsen classes in one cusp orbit
 with those in another cusp orbit by intersecting the entries in one orbit with the shift applied to the entries in the other orbit. 

Its blocks correspond to the reduced Hurwitz space components, and the matrix of the pairing is symmetric if <i>r</i>=4. The matrix gives much other information, too. For example, when <i>r</i>=4 you can read of the genuses of the components from it. This invariant of the Hurwitz moduli space, attached to a Nielsen class, greatly simplifies the combinatorial data about the Hurwitz space. Further, it is compatible with computing with projective sequences of Nielsen classes, like those arising in Modular Towers.