If two gyroscopes were to be started with parallel angular momentums, linked by a rotating tube (one perpendicular to the axle), then a second pair added that is the exact reverse such that it has opposite gyroscopic precession, the two linked such that they may not move relative to each other, and then each gyroscope moved by the rotating tube in the opposite direction of the one on the opposite end for each pair, where would the energy go that was used to force the gyroscopes to move away from their original orientation? I can add a drawing of the apparatus if it would help.
In general, energy is not required to change the direction a gyroscope is rotating. Similarly, it doesn't require any energy to redirect an object with linear momentum either (we don't consider there to be energy transfer when a billiard ball caroms off a cushion).
In the case of the bouncing ball, there's no energy transfer because we consider the wall/cushion to be fixed in place. So the force applied happens over zero distance.
In the case of the gyroscope, the gyro precesses at right angles to the applied force. Again, the cross product of the force and the distance moved is zero.
You raised a question in comments about a spring pushing on a gyroscope. I don't think I understood the geometry. A diagram might help.
Your diagram shows a spring pulling down, so there is force beyond just the weight of the gyroscope. We could do the same thing by adding a weight to the unsupported end of the gyroscope.
So the statement that it doesn't take energy to move the axis is correct. But it does take a bit of energy to start moving it. (just like it takes some energy to accelerate a ball, but none to keep it rolling, sans friction). This energy comes from a slight drop in the gyroscope.
The gyroscope falls slightly. As it falls, it picks up speed sideways. The greater the downward force, the faster it moves sideways. The faster it moves sideways, the more it counters the downward force until it reaches an equilibrium. Lowering the force would reduce the sideways speed.
This is quite old question, but recently I came across a problem which was describing a setup with multiple gyroscopes that counteract each other and a question of how energy would transfer.
The accepted answer is correct, however counterintuitive it might be. I would like to expand upon it by providing a bit of high level explanation and visual demonstration.
You see, precession and nutation are vital parts of gyroscopic effect, and for them to work, the gyroscope needs to be able to rotate freely in the space. In your setup gyroscopes are rigidly connected to each other and their precession cancel each other out. This would be effectively the same as removing one of degrees of gyroscope's freedom (specifically the one used for precession movement). This leads to gyroscope losing its properties and acting basically the same way as if it wasn't spinning at all.
You may look at it as if you had a force, that was acting directly against precession direction. Such force would have its own momentum, and if you do vector product, this momentum would be aligned with the original force, that caused precession in the first place.
I found an old Soviet demonstration video that clearly shows this effect: https://www.youtube.com/watch?v=j7IdaK-rERI
At first, the gyroscope's base is locked and can't rotate. When the scientist pushes on the ends of gyroscope axis, it rotates freely with little resistance and slows down due to friction. Then the base is unlocked and suddenly the gyroscope shows strong resistance when attempting to changes its orientation. You may also notice, that in the first half some attempts to push the gyroscope actually DO show resistance and others seem completely effortless. If you look closely, this is because the base is not completely locked and has some slack, which gives the gyro a bit of room for precession, which causes this small bit of resistance.
I hope this will help those who may be interested in this small but very important aspect of gyroscope behavior, which is not discussed in lot of details for some reason