Why does the liquid in an Oralmedic stick flow uphill after I turn it upside-down?
The most likely explanation is a combination of two capillary effects, where a combination of surface tension and adhesion of the liquid to surrounding solid surfaces causes macroscopic flow to occur.

For the first, the internal diameter of the hollow stick is small enough that if one were to pull off the cotton buds at the top and bottom, capillary forces would likely cause the fluid to flow only very slowly through it. This provides some resistance to flow, sufficient to keep the liquid from gushing up to the 'snapped bud' end when you invert the stick. The surface tension associated with this capillary effect also results in the liquid behaving like a "plug" in the hollow tube: gas cannot "slide" around the liquid meniscus, which is also an important component to the physics.

The other role of capillary action is active in the fine threads of the lower cotton bud itself.  Once the top bud is snapped, the liquid starts to slowly flow down toward that unsnapped bud. Once the liquid reaches it, it wicks aggressively into the finely structured cotton.  Since no gas can get past the liquid in the tube, conservation of mass requires that the rest of the liquid continue to flow down toward the unsnapped bud.

The reason why flow continues toward the unsnapped bud when you invert the stick is a combination of all three of the above factors: the capillary action drawing the liquid into the unsnapped bud is essentially unaffected by the inversion of the stick, and since no gas can slip around the remaining liquid in the tube, that remaining liquid has to continue flowing up toward the unsnapped bud.  

Another way to think about it is this: as each microscopic parcel of liquid wicks into the unsnapped bud, it leaves a region of slightly lower pressure behind it.  Since the end of the liquid column facing the snapped bud is open to atmosphere, that results in a pressure differential against the snapped-bud side of the liquid column that pushes the liquid up toward the unsnapped bud.