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There are little bits of xantham gum in the liquid of the drink. Shaking it moves the particles around, but the end distribution is similar to the distribution before the bottle was shaken. Rotating it initially doesn't really move the particles (relative to the bottle) until the rotation is stopped, at which point the particles continue to spin as if the bottle were still moving.
When a solid is more dense than the liquid it is placed in, the solid sinks. When a solid is less dense than the liquid it is placed in, the solid sinks. This is a rare case when the solid and liquid are almost equally dense.
The xantham gum bits and the corn syrup liquid have virtually the same density (1.052 g/cm3). This enables the xantham gum to float in equilibrium with the liquid. The equilibrium causes a random distribution of xantham gum whenever the bottle is shaken.
Spinning the bottle spins the entirety of the liquid inside, so the xantham gum appears to spin with the bottle. When the bottle stops spinning, the liquid continues to flow, and the xantham gum, as a result, continues to spin.
It should be noted that, despite the nearly identical densities of the solid and liquid, a settling of some kind should occur over a long enough time period. It doesn't. The reason for this observation is one of the drink's ingredients: gellan gum. This polysaccharide forms a fragile, net-like matrix that lightly holds the bits of xantham gum in place, aiding in the gravity defiance witnessed.
This material is based upon work supported by the National Science Foundation under Grant Number 1246120