  This paper introduces the $J^{2+}$-invariant for pairs of generic immersions.
The new invariant behaves like Arnold's $J^{+}$-invariant for generic
immersions as it is invariant under homotopy including inverse tangencies and
traversing triple points, but changes by traversing direct tangencies. It has
several other properties, for example the invariant for pairs is completely
independent of the $J^{+}$-invariants of the single immersions, which we will
prove in Theorem B.
