I've wondered about this for ages. If we create a pair of flywheels that rotate in the opposite direction with the same angular momentum, but are co-located and have the same mass and inertial moment (one can imagine various ways to accomplish this, at least approximately) -- it is clear to me that there cannot be any precession forces, but, if we try to rotate the entire assembly around an axis perpendicular to the axis of flywheel rotation, will the force needed to produce this secondary rotation be the same as if the flywheels were stationary, or will it require a proportionally greater force to rotate in this way, as it does with a single flywheel?
two concentric and counterrotating flywheels preclude all precession forces regardless of which plane the axis is rotated in. this is assuming the connection between the two flywheels is sufficiently strong--it make break from tension/compression due to each flywheel experiencing its own forces. refer to the diagram i just drew up.
the black rectangles are the two flywheels, the connecting line is the physical connection and also the axle which both flywheels are concentric. the red arrows show the direction of angular momentum (along x axis), while the red circles indicate the direction of rotation (around x axis).
the blue arrows indicate the precession forces experienced by both flywheels when the whole system is rotated in the direction indicated by the curved blue arrow. this is what causes tension/compression in the connecting bar, but otherwise zero torque on the system as a whole.
the green arrows indicate the same forces if the system was rotated the other way (counter to the blue curved arrow).
the situation is similar for rotation of the system in any other plane.