Imagine you have a gyroscope that has two spinning parts, one on top of the other. When you pull the string, the two halves spin in opposite directions. Would this cancel out the gyroscopic properties, or double them?
Short answer: it cancels the gyroscopic effect (with caveats).
As long as the system holds together (see below), if the two halves spin with exactly the same magnitude but opposite sign angular momentum, from the point of view of an outside observer, the system behaves like one of zero angular momentum. In particular, it takes negligible torque on the part of an outside observer to rotate the system, and there is no phenomenon of precession or nutation. Indeed this kind of principle is sometimes used in robotics and mechanical engineering to allow high speed rotating components to be manipulated easily.
However: from the standpoint of each rotating component, each requires a torque to change its own angular momentum. Indeed, if you spin the system quickly, you're forcing the angular momentum of the two separate components to change extremely fast. The two spinning components must therefore exert huge torques on one another to achieve this. Rotation of the whole system, although easy for the outside observer, begets huge stresses on the shaft joining the two components. If you set a system like this up and rotate it, you can see the shaft between the components bending slightly at right angles to the plane of rotation, as the massive torque between the components sets up high bending moments in the shaft. This kind of experiment needs to be done with great care, with very lightweight components and with safety glasses on. Systems like this can explode if the joining shaft fails, and whenever the principle is exploited in robotics, the control system imposes very strict limits on the maximum rate of rotation of the system as a whole, if the rotation is in a different plane from that of the two components' angular momentums.
This would cancel out the gyroscopic properties. Here is a youtube video that demonstrates this effect: https://www.youtube.com/watch?v=bzbVwiIeM0M