It's called a power-back as apposed to a push-back. The DC9 can do either.

In order to do power-backs the airline must have authorization for power-back operations in their FAA operation specifications (OpSpecs). The OpSpecs will list the exact airports and gates at which a particular type airplane is authorized for power-back operations. Not all gates at all airports will be authorized.

Wing walkers are used and a marshaller gives the Captain hand signals during the power-back. It starts out with powering forward a few feet to get the airplane moving. Once the airplane starts rolling forward the reversers are engaged and the airplane powers-back. The Captain keeps his feet on the floor during the power-back as you can't apply brakes while backing up as doing so could cause the plane to tip back onto it's tail. To stop the power-back the reversers are stowed and forward thrust is used to stop the airplane.

All airplanes are more prone to FOD when in reverse at slow speeds. The reversers work by redirecting the exhaust somewhat forward but a lot of the exhaust goes up, down and to the sides as well. This will kick up any debris on the ground and could easily be ingested into the engine. During landing, the airplane is moving fast enough that the airplane is well down the runway by the time any debris would reach the engine's height but you do generally stow the reversers by 60 knots, or so, to reduce the chance of FOD ingestion at slower speeds--this is especially true with under-wing mounted engines.

Not all airlines have authorization for power-backs so while you see them frequently on AAL MD80s there will be other MD80/DC9 operators who never use them.