Is quick left brain vs right brain test fake? When I saw it at first she did rotate to the right. 

With some mental effort I could make her rotate either way in half rotations where it looks like she dances not rotating in my minds eye, but I can only hold that until she starts to rotate either way again. After a few minutes it looked like her leg twisted one way and her body the other. 

What would a person see if their brain is equally balance without a dominant side? What does it say when a person has the inability to only see a rotation? 
Does this test have any basis if you are left or right brained? Does switching the direction from mental practice of build any part of your brain when you get to a point where you can do it more freely?

Does this picture identify anything about cognitive abilities?
Without references, you state:


  In this picture if you noticed her rotating clockwise you are right
  brain dominant.


We discourage asking questions on this site (or in science in general) which are based on unverified/referenced premises. However, given that this is a very common notion found online, I will try to answer your question by first helping you rephrase it.

What you should have asked is something along the lines of:


  I have read online articles (e.g., on learning-mind.com) which claim that if you perceive the
  figure in the following image rotate clockwise it means you are right
  brain dominant and when you see her rotating counterclockwise it means
  you are left brain dominant. However, the mechanism behind this is not
  well explained. How does this test work and what exactly does it imply?


To answer:

This figure is known as the 'Spinning Dancer' and there is a wikipedia article on it. The wikipedia entry states that:


  In popular psychology, the illusion has been incorrectly[6] identified
  as a personality test that supposedly reveals which hemisphere of the
  brain is dominant in the observer.


They link to NEUROLOGICAblog, "Left Brain – Right brain and the Spinning Girl", which explains that:


  This news article, like many others, ignores the true source of this
  optical illusion and instead claims it is a quick test to see if you
  use more of your right brain or left brain. This is utter nonsense,
  but the “right-brain/left brain” thing is in the public consciousness
  and won’t be going away anytime soon.


Instead, they explain what is actually going on:


  The spinning girl is a form of the more general spinning silhouette
  illusion. The image is not objectively “spinning” in one direction or
  the other. It is a two-dimensional image that is simply shifting back
  and forth. But our brains did not evolve to interpret two-dimensional
  representations of the world but the actual three-dimensional world.
  So our visual processing assumes we are looking at a 3-D image and is
  uses clues to interpret it as such. Or, without adequate clues it may
  just arbitrarily decide a best fit – spinning clockwise or
  counterclockwise. And once this fit is chosen, the illusion is
  complete – we see a 3-D spinning image.


In case you have specific questions on what 'hemispheric dominance' is and how we can determine it, the article (to my understanding, correctly) mentions:


  We also do have hemispheric dominance, but that determines mostly your
  handedness and the probability of language being on the right or the
  left. There is also often asymmetry for memory, with some being right
  or left hemisphere dominant.


I'm certain you can find more scientific sources on this by googling (I would recommend Google Scholar), or by looking on this site. At the time of writing, we have 12 search results on 'hemispheric dominance'.