This is neat. I think that {left, right, neither} is too broad a categorization, and suffers from the usual problem of self-reporting: most people are bad at judging themselves, and this is before you factor in the desire to paint themselves in a better light. An open affiliation could also make some people reluctant to participate (particularly with point (3) in your top-level post, seeing the distribution of posts upvoted).

Instead of having users explicitly choose from {left, right, neither} I would suggest not asking at all, and using analysis to find groups that vote the same way. You don't even need any labeling, just scores for how much the article agrees with the user's previous voting pattern and how broad of a group the article appeals to.

Now that I'm done insisting that the bike shed be painted blue I hope this takes off -- echo chambers do no-one good.