Lots of interesting theories here, and some contributions by people who know what they are writing about.

Many types of jet airliners are equipped with thrust reversers, though certainly not all are. As a general rule, jet airliners with the engines mounted on/in the tail cone can be powered back (backed up using the thrust reversers) with reasonable ease and safety, though lookouts are required all around the airplane to make sure the airplane stays out of the way of other aircraft and doesn't back into or over something.

Back in the days when Boeing 727s were in major airline service, "powering back" was not an uncommon sight. There are still some hazards involved, as you stir up a lot of trash and junk, and you can damage the landing gear and some airframe parts by turning too sharply.

Airplanes with jet engines mounted under the wings cannot be powered back. Or at least they should not ever be. It is done, rarely, but it is dangerous and can be damaging. Most companies forbid it.

Airplanes, either turbine or piston, that have reversible propellers can be backed under power much more easily and safely. The C-130 is a good example. Because it is a high-wing airplane, the propellers don't stir up near as much trash as they would in the same maneuver with a low-wing airplane.

We used to back our piston engine Convairs and Douglases all the time. It isn't too hazardous if the ramp is kept clean. I once backed a DC-6 into a hangar with about nine lookouts. I had less than eight feet clearance on each wingtip, and, of course I could not see where the bejezus I was going. So I trusted the guys on the ground, and we backed the airplane almost 300 feet without hurting anything.

I would not want to have tried the same thing with the 747s I later flew. And of course you don't run jet engines in hangars, for obvious reasons.

There is another way to back an airplane. Almost any multi-engine airplane can move backward by "wanking." You set the brakes on one side and power up the engine on the same side. That swings the airplane around, putting the opposite main wheels further back. Repeat the process on the other side, and so on. It's hard on the airplane, and, again, you can't see where you are going. But it can be done in a pinch.
Source(s):
retired Boeing 747 Captain
also Boeing 727
Douglas DC-4, DC-6, C-54, C-118--Convair 240, 440, 580