*cough*
http://en.wikipedia.org/wiki/Radiation_pressure
If radiation can exert pressure, as explained by my previous link, then the pressure MUST be exerted in the opposite direction that the radiation is moving. That is simple Newtonian physics. So if the photons exit the flashlight in one direction -- AFTER BOUNCING OFF THE REFLECTOR AND EXERTING PRESSURE ON IT -- then the flashlight must move in the opposite direction.
Photons have no mass, but photons don't
need mass in order to exert pressure. Photons contain energy, and a certain amount of energy can always be mathematically equated to a certain amount of momentum. Quantum physics is confusing at times, but basically, quantum particles are capable of behaving like all of their mathematical equivalents at once. That means photons can bang into things and move them even though they have no mass, because their momentum is still absorbed by the object they collide with. Older versions of physics didn't predict this because older physicists had no experimental knowledge of quantum particles, and they were unable to discover and measure quantum effects because they didn't have equipment sensitive enough.
Notably, lasers are used in experiments with antimatter to pin the antimatter particles in-place so they can be observed. This works because the photons do exert pressure, and because they are laser photons, they all travel in the same direction with the same energy so the amount of pressure can be very precisely controlled.