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.
What you have to worry about most when you reverse direction of a motor, is that you do not put too much current into either the motor or the electronics/switches that control it. 

When you connect a voltage source across a motor that is at rest and either has a large inertia or a locked rotor, you get a large current flowing through it = V / R where R is the stator winding resistance of the motor. This is called the stall current.

If you are running at full speed with a voltage source across a motor, and you immediately reverse the polarity of the voltage source, you can get up to 2x the stall current, because the voltage source is then at the opposite polarity of the motor's back-emf. This can be too much current, and if that's the case then you have to control the rate at which you reverse voltage across the motor, by using PWM or some other way besides a hard voltage reversal.