Moment of Inertia (MOI) is typically used to describe a rigid body, so I don't know how well I can express my question in those terms. However, it seems like the best framework for getting at my issue.
One description of MOI is that it determines how much effort it would take for me to rotate an object. For a gyroscope with no gimbals (so, more of flywheel mounted inside a frame), the mass distribution for the object is the same when the flywheel spins as when it is not spinning. However, there is an additional angular momentum vector in the system when the flywheel spins. The MOI has no place for added angular momentum, but the addition of angular momentum would mean that the torque that I apply to the object to rotate it would result in an orientation other than what I intended.
This means (or seems to mean) that the torque required to get the object from orientation A to orientation B is different in direction, but not in magnitude, when the flywheel spins compared to when it does not spin.
Thus, it seems the addition of angular momentum reorients the MOI, at least in a practical sense, without changing its values.
Is this correct? And if so, are there formulas for determining this reorientation for given values of the MOI and angular momentum?