What determines the spiral direction of plants?
First of all, you assumption seems to be incorrect. Plants in same part of the world can twine in opposite directions.

Little seems to be known exactly. Gravity has nothing to do with plant's chirality (or "handedness") or spiraling:


  How plants do this is still unknown. Darwin proposed that it was "autonomously induced." This came to be known as the internal oscillator theory. In the 1960's the gravitropic overshoot theory explained mathematically how circumnutation could spontaneously occur as a result of forces related to gravity and growth. Recent experiments on the Space Shuttle have demonstrated that gravity is not necessary for circumnutation.


Here is another link to a blog, but it does contain reference to original paper on observation of chirality. Main conclusion, taken from the abstract, is:


  The predominance of right-handed helical growth in climbing plants cannot be explained by hypotheses attempting to link plant growth behaviour and global location. One alternative hypothesis for our findings is that the widespread phenomenon of anticlockwise twining arises as a function of microtubule orientation operating at a subcellular level.


To elaborate a little on that, remember that actin filaments as well as microtubules are all chiral polymers. That is, they form spiral structures with defined orientation. Even more, microtubules all start roughly from the nucleus away to membrane, so this might suggest cellular mechanism for preference toward twining chirality. However, this does not address your observation that given plant changes direction of growth twice.