I'm not sure if this will help, but you could try to imagine a moment in time where the gyro is horizontal with a vertical axis, and that a horizontal plane is used as a reference for the gyro at this moment in time. Assume the gyroscope is precessing along an east - west axis, so that the north side of the gyro is precessing "upwards" and the south side is precessing "downwards". For any point on the gyro, the acceleration component perpendicular to the plane is greatest on the east and west side, and zero on the north and south side (the acceleration component would be a direct reaction to input torque). For any point on the gyro, the velocity component perpendicular to the plane is most negative on the south side, and most positive on the north side, and zero on the east and west side (the velocity component would correspond to the gyro's precession reaction).
The precession reaction is 90° out of phase "behind" the applied torque. Helicopters are affected by this and the cyclic (pitch and roll) output is adjusted 90° "ahead" of the control input to compensate.
This description is ignoring the fact that the overall angular momentum of a gyro is affected by the rate of precession. The actual precession reaction is slightly different than the idealized one.