Why does spin recovery involve cutting the throttle? When recovering from a spin, the first step is generally to cut the throttle:


  ... Otherwise, to recover from an upright spin, the following generic procedure may be used: Power is first reduced to idle and the ailerons are neutralized. Then, full opposite rudder (that is, against the yaw) is added and held to counteract the spin rotation, and the elevator control is moved briskly forward to reduce the angle of attack below the critical angle. Depending on the airplane and type of spin, the elevator action could be a minimal input before rotation ceases, or in other cases the pilot may have to move the elevator control to its full forward position to effect recovery from the upright spin. Once the rotation has stopped, the rudder must be neutralized and the airplane returned to level flight. This procedure is sometimes called PARE, for Power idle, Ailerons neutral, Rudder opposite the spin and held, and Elevator through neutral. ...  [Wikipedia]


This is extremely counterintuitive; given that another vital step in spin recovery is to apply full down elevator1 to get the nose down, one would expect that the blowback from the engine (especially for tractor-propeller aircraft) would aid in recovery by striking the underside2 of the deflected elevator, and, thereby, pushing the tail up (and the nose down), making it seem, at first glance, that it would be best to firewall the throttle in order to get the tail up and the nose down as quickly as possible.

In addition, even without the contribution from the elevator, applying full throttle would:


drag the aircraft forwards through the air, increasing the forward component of its motion and decreasing the angle of attack seen by all the various parts of the aircraft;
for tractor-propeller aircraft, blow air at high speed backwards over the wings, dramatically lowering the local angle of attack felt by the wings.


Both of these effects would tend to destall the aircraft and transform the spin into a boring old trivial-to-recover-from spiral dive.

So why do spin-recovery procedures for most aircraft generally call for cutting the throttle, rather than firewalling it?



1: Or, if spinning while inverted, full up elevator.

2: Replace "underside" with "upper side" if spinning inverted.
Applying engine power can indeed help to end a spin. But the added thrust will make an overspeed in the ensuing spin recovery much more likely. Also, applying asymmetric thrust on the wrong side will flatten and stabilize a spin. These are the main reasons why the engine should be idle in spin recoveries.
Remember the Boeing 307? The first passenger airliner with a pressurized cabin and the civilian version of the B-17. It was test-flown by KLM personell on March 18, 1939. It entered a spin after a stall and the rudder became stuck in spin direction. The pilot applied asymmetric power in order to stop the yawing motion and succeeded in doing so, but in the subsequent recovery the aircraft oversped and the outer wings broke off. Here is the official accident report.

Boeing 307 in flight (picture source). Note the small fixed vertical surface which was replaced after the crash with a much bigger one.
Symmetrical thrust can be helpful or adverse for spin recovery, depending on circumstances. NASA TN D-6575 from December 1971 says that

no systematic study of the effects of applying power during a spin

is known and

The effect of applying symmetrical power in a spin is believed to be insignificant. A number of varied observations have been made through the years by many people, and the conclusions regarding power effects on spins range from favorable to adverse. […] In almost all cases, the results were not obtained under controlled conditions. Therefore, the type of spin, the center of gravity, the angle of attack, the spin rate, and the line of thrust with respect to the center of gravity were not identified.

Therefore, a general recommendation for applying symmetric thrust for spin recovery cannot be made and only asymmetric thrust has shown to be helpful if applied correctly:

the application of thrust had no effect unless the thrust axis was displaced from the center of gravity and thereby produced a moment.
[…]
For asymmetric power for a twin-engine configuration with the engines mounted on the wings, power from only one engine can produce a large asymmetric yawing moment, which will be favorable or adverse to the spin and recovery, depending on the direction of the moment. Both model and full-scale spin-test results of multiengine airplane designs have shown that power on the outboard engine (e.g., the right engine in a left
spin) can create a large prospin yawing moment, which can cause a flatter and faster spin. On the other hand, power on the inboard engine can create an antispin moment to aid spin recovery. Normally, the manipulation of thrust can be confusing and disastrous if the power is applied to the wrong engine. Therefore, unless asymmetric power is necessary to aid recovery, it is generally recommended that for a multiengine airplane, the throttle be retarded to the idle position on all engines during a spin.

It should also be mentioned that the extreme angle of attack at the inboard engine will produce an extreme sideways displacement of the thrust vector on a propeller or uneven flow in a jet intake. In both cases such a flow condition was most likely not anticipated during design and could result in the loss of the propeller rsp. a compressor stall in the jet engine if full thrust is applied.
While I do not know of a propeller having been lost in a spin, during the flight test of the Dornier 328 an extreme sideslip did indeed overload the left propeller, parts of which tore a massive hole into the fuselage.