It's not moving the HD that causes precession like Splinter says. Some Googling gave me this "definition" of gyroscopic precession:
If you have a spinning gyroscope and you try to rotate its spin axis, the gyroscope will instead try to rotate about an axis at right angles to your force axis.
Note the word rotate in the italic text. If you have a force that tries to
translate the gyroscope, then it will just move in the direction you want. No precession at all.
Gravity is a translational force. Gravity in itself can't cause gyroscopic precession. In all the examples and formula's where there is talk about gravity causing this, it is actually the unevenly applied normal force that is the cause. Take the bicycle wheel in
this example. In
Figure 1 the axis is supported on both sides, so the gravitational force pulls down and the normal force on both sides of the axis pulls up. The result is that nothing moves, as the forces cancel each other out. In
Figure 2 and 3 one of the supports is removed. The gravitational force still tries to pull the whole wheel down, but there is only a normal force on one side of the axis pulling it up. The result is that the wheel starts to precess.
Now back to HD's. The last time I looked inside a HD, the axis of the platters was only attached to something on the bottom side of the disk. So it looks like we can use the model with the uneven normal force. The axis of the platters isn't free to move in any direction it wants. It is fixed by bearings, but these have some give (eng?). As far as I can follow
xilencer's arguments, he states that the gyroscopic precession causes the axis of the platters to tilt as far as the bearings allow it. What I don't understand is, why he thinks this will only happen when the disk is mounted at an angle. Seeing all the examples, this will also happen when the drive is mounted in a vertical position. Moreso when the drive is vertical, this effect will be the strongest, as the torque will never be bigger than in that orientation.
Conclusion:
Whatever the reason some manufacturers say drives can't be mounted at an angle, it cannot be the effect of gyroscopic precession on the platters, because they all allow us to mount the drives vertically.