Personally for me this is the easiest method, I would split the source geometry into smaller segments so that you can more finely control where the Material ID changes.
However I've also attached a version that uses the mapping coordinates as you suggest. There are a few complexities with this, as you've noticed the red/white pattern is reversed on each side. This is because in most cases you wouldn't want the texture flipped on the U axis on the reverse, however in this case it prevents the style from working correctly. To remedy this I have split the base geometry into 2 material IDs, the original and one where the texture has been rotated 180 degrees. You can then apply this to the reverse so that the colours match on both sides.
I hope that helps, please let me know if you have any further questions.