Just like the reaction of the weight of a body is a force acting on Earth towards the body, where and in what direction does the reaction of a gyroscopic force act?
There is no "gyroscopic force". The best you can say is that there is a "reaction torque" when you apply a torque to a gyroscope. When a torque (which is a vector $\vec\Gamma$) is applied to a gyroscope with angular momentum (also a vector $\vec L$) the resulting motion of the gyroscope is a precession - the tip of the angular momentum vector rotates along the direction of the torque vector.
If you attempt to move a gyroscope such that its axis of rotation has to change, the reaction will be opposite to the direction of the torque you apply.