Solar system Centaurs originate in transneptunian space from where planet orbit crossing events inject their orbits inside the giant planets' domain. Here, we examine this injection process in the three-body problem by studying the orbital evolution of transneptunian asteroids located at Neptune's collision singularity as a function of the Tisserand invariant, T. Two injection modes are found, one for T>0.1, or equivalently prograde inclinations far from the planet, where unstable motion dominates injection, and another for T<= 0.1, or equivalently polar and retrograde inclinations far from the planet, where stable motion dominates injection. The injection modes are independent of the initial semi-major axis and the dynamical time at the collision singularity. The simulations uncovered a region in the polar corridor where the dynamical time exceeds the solar system's age suggesting the possibility of long-lived primordial polar transneptunian reservoirs that supply Centaurs to the giant planets' domain.