  This paper introduces the $J^{2+}$-invariant for oriented pairs of generic
immersions. This invariant behaves like Arnold's $J^+$-invariant for generic
immersions as it is invariant when going through inverse tangencies and triple
points, but changes when traversing direct tangencies. It has several useful
properties, for example its independence of the $J^{2+}$-invariants of the
single immersions forming the pair. Also it is invariant under simultaneous
orientation change. Therefore, one can define two $J^{2+}$-invariants for each
pair depending on its orientation, those two invariants are not independent
from each other. Furthermore, the invariant is extended to $J^{n+}$ for links
of $n$ oriented immersions.
