No they don't. Any spinning wheel will create a force that tries to preserve its axis of rotation. If you have two counter-spinning wheels that share the same axis and try to tilt them, they will both try to preserve the same axis of rotation and thus exert a force in the same direction.
The result was that counter rotating gyroscopes had no net effect.
The only thing that matters is total angular momentum.
The resistance to being tilted is a _secondary_ effect. The push gets translated into motion at right angles to the push, and this motion gets translated into push at one more right-angle, exactly countering the original push. For two counter-rotating gyroscopes, this first-stage motion of one is exactly countered by the other, leading to no net effect. Really, truly.
