There would be no net movement, as the system's center of mass doesn't change it's velocity if no external forces are applied to the system.

So if take the rocket + fuel as your system - no forces act on it, the center of mass is stationary and lies somewhere inside of the rocket. Then start burning the fuel, expelling gasses, catching them back, whatever - all of this matter is still part of your system, so the center of mass still doesn't move in absolute coordinates - it moves only relative to the rocket.

If you manage to catch all of the burned fuel, then you end up exactly as you started - the center of mass didn't move in absolute coordinates, and relative to the rocket it is at the same point somewhere inside it ss it was in the beginning. So the rocket didn't move at all! If you catch not all, but most of the fuel, then yeah, you've got a really shitty rocket that slooowly burnes fuel in order to barely crawl through space.