newtonian mechanics - What determines the direction of precession of a gyroscope?

I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the gyroscope.

However, the direction, either clockwise or counterclockwise, of this precession seems to me a bit arbitrary beyond mathematical definitions. (What if torque were defined by a "left hand rule", for example?).

What's the fundamental physical reason that the precession of a gyroscope would be directed in a certain direction?

Related question.