The most stable axis of rotation for an object is the one with largest (moment of) inertia. Although the axis with lowest inertia is also stable, if there is energy loss (e.g. air drag) it will eventually end up rotating along the maximum inertia axis.
Nice description of dissipation-induced instability. Famous example: Explorer I, launched in 1958 to become the 1st US spacecraft to achieve orbit.
Plan A was to stabilize the attitude of this long, narrow rocket-shaped satellite by spinning it about its centerline. Would've worked in the absence of dissipation, as this was its axis of
minimum moment of inertia. But by the end of its 1st orbit, Explorer I was no longer spinning like a bullet. Instead, it was spinning like a propeller about its axis of
maximum moment of inertia, just as you described.
One teensy dissipation had been overlooked: Elastic heating of the 4 whip antennas as they flapped after release. Ultimately, the heat lost to space through the antennas took only a tiny bite out of the spacecraft's rotational kinetic energy. But that was all it took to switch Explorer I from bullet to propeller mode. Like the ISS T-handle, the spacecraft's total angular momentum changed very little in the mode switch. It just took on a different outward form.
An outside engineering professor familiar with this kind of instability tried to warn NASA months before launch, but security measures kept the heads-up from reaching project engineers. Seven months after launch, he published a paper spelling out the cause of the mode switch. Only then did the Explorer team tumble to what had happened.