What causes the water in this fountain to reverse direction?

This is probably caused by a ring-shaped whirl of air (similar to a "smoke ring") under the water sheet which is driven by the speed of the water. This whirl flow produces a lower air pressure at the inner side of the falling water sheet that sucks it to the inside.

It's called a "water bell". It's produced by a balance of inertia and surface tension (along with gravity). See this link for a little more detail, or this one for a more thorough analysis.

Yes, this effect comes from fluid dynamics. The material of effect is pressurized air which is circle-flowing under the water. At low velocity, the air is not having large impact. As velocity gets higher, air seeks to fill the low-pressurized region in lower segment of flow, creating more tight loop. This loop is sucking water to closure it, also protecting from mixing with outside air.