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 deg) state. We estimate that at any given time, there should be ~1 kilometer-scale or larger escaped asteroid in a transient direct (prograde) orbit with semimajor axis near that of Jupiter’s (a~aJ), 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 about 1% of the transient asteroid co-orbital population is on retrograde orbits with Jupiter. This population, like the recently identified asteroid 514107 Ka'epaoka'awela (2015 BZ509), can spend millions of years with a~aJ including tens or hundreds of thousands of years formally in the retrograde 1:-1 co-orbital resonance. We compare the production of jovian co-orbitals from escaping near-Earth asteroids (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. Escaping NEAs are thus likely the precursors to the handful of known high-inclination objects with a~aJ. We postulate that the primordial elimination of the Solar System’s 3-10 AU planetesimal population could provide a supply route to create either a metastable reservoir or just eroding remnant of a “polar corridor” in the outer Solar System for the high-inclination Centaurs, using the demonstrated existing path to such orbits (Greenstreet et al 2012, ApJ Letters, 749, 39) via our already known most-massive planet, Jupiter!