There are many questions about gyroscopes, and this one: (Confusion about what happens when the rotating axis of a gyroscope is rotated) remind me of an odd behaviour of a toy gyroscope that I had.
It is made in a way that besides the rotor, also the case spins when placed on the ground. So, I think it is not same situation described by that question.
The expected behaviour is an increased precession as the system loses energy until falls down. And it happened many times indeed.
But sometimes the outcome was different: after some time of precession, its axis changed from a tilted to a vertical direction, stayed there for some time (without precession) until the angular velocity decreased too much and it fell down.
I made a video of one of that events: (https://www.youtube.com/watch?v=UZYlW6ErqdI)
May be the existence of 2 rotating coupled bodies is the reason, but I don't know how it could lead to such a counter intuitive movement. It seems that some of the lost kinetic energy is transformed in potential, raising a little bit the centre of mass of the device.