We demonstrate dynamical pathways from main-belt asteroid and Centaur orbits to those in co-orbital motion with Jupiter, including the retrograde (inclination $i>90^o$) state. We estimate that at any given time, there should be $sim1$ kilometer-scale or larger escaped asteroid in a transient direct (prograde) orbit with semimajor axis near that of Jupiters ($asimeq a_J$), with proportionally more smaller objects as determined by their size distribution. Most of these objects would be in the horseshoe dynamical state, which are hard to detect due to their moderate eccentricities (spending most of their time beyond 5 AU) and longitudes relative to Jupiter being spread nearly all over the sky. We also show that $approx$1% of the transient asteroid co-orbital population is on retrograde orbits with Jupiter. This population, like the recently identified asteroid (514107) 2015 BZ$_{509}$, can spend millions of years with $asimeq a_J$ including tens or hundreds of thousands of years formally in the retrograde 1:-1 co-orbital resonance. Escaping near-Earth asteroids (NEAs) are thus likely the precursors to the handful of known high-inclination objects with $asimeq a_J$. We compare the production of jovian co-orbitals from escaping NEAs with those from incoming Centaurs. We find that temporary direct co-orbitals are likely dominated by Centaur capture, but we only find production of (temporary) retrograde jovian co-orbitals (including very long-lived ones) from the NEA source. We postulate that the primordial elimination of the inner Solar Systems planetesimal population could provide a supply route for a metastable outer Solar System reservoir for the high-inclination Centaurs.
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