Question: Can you run a [BLDC motor](http://en.wikipedia.org/wiki/Brushless_DC_electric_motor) backwards without damage?

Is it OK to drive a model airplane BLDC engine backwards while landing, so it gets a little "reverse thrust" and come to a stop a little quicker on the runway?

Is it OK to drive a model helicopter BLDC motors backwards so it can hover upside-down?

Or do I need to design the hardware so that it never drives the motor backwards, under any circumstances, no matter what the pilot on the ground does at the transmitter?

Answer: 
BLDC motors usually just use permanent magnets on the rotor (be it in-runner or out-runner) and use a set of windings on the stator connected in a three-phase delta or wye configuration. The speed controller just generates a variable-frequency, three phase waveform to power the motor. Since the windings are symmetric, electrically there's no reason you can't turn the motor in either direction.

As for whether it's a good idea to run a prop backwards on landing, that's more of an aeronautics problem than anything inherently electronic. Having flown some r/c planes, it seems to make sense to me that if you reverse the prop on landing, you're basically just applying a braking force along the line of the axis of rotation. If that line passes above the center of gravity (not below it), that should torque things so that the tail will stay down, so you should be stable if that's the case. If the prop axis is below the CG, though, you're looking at forward torques that would drive the nose down, which would result in damage.
Thrust on helycopters is controlled by varying the propeller pitch, not the motor speed/direction. In an helycopter, the main rotor motor turns almost always at the same speed. 

Inverted hovering needs a special designed [swash plate](http://en.wikipedia.org/wiki/Swashplate_%28helicopter%29) that allows positive and negative blade pitch. 

For the original question, yes, you can drive a brushless dc motor in both direction. 
Stopping it quickly (with a propeller connected to it), change direcion to the thrust, and keeping your plane on the track, is another story :-)