With a toy gyroscope, you can demonstrate to yourself some of the YoYo characteristics described here. But watch the YoYo as it rotates its position to the left if it is rubbing the left rim, and to the right if it is rubbing the right rim. The vertical string, in contact with the spinning YoYo at the top of the rim applies a force that tries to bring the YoYo to the vertical position.
Gyroscopes resist attempts to change the angular position of the axis of the spin.
If your YoYo is not well balanced, you may see some slow precession. Wood YoYos often have one side heavier than the other. If you have one like that, you might be able to guess which side is the heavier by the direction of precession you observe. The twist in the vertical string in combination with the weight of the YoYo can impart a slight torque on the axle and cause a slight and slow precession. YoYos use fine sandpaper to remove enamel at the rim and to smooth the wood at the rim, then polish the wood at the rim with a hard wax, leaving a surface shine and no excess wax.