I'm sure some smart people here have some reasons why not. But here's my take: It would be a rhetorical advantage to claim that a left-right political model only describes a capitalist democracy. Because we're neither capitalists nor democrats then it simply doesn't apply to us.
We're not left or right. The qualities of socialism have conservatism and liberalism mixed in because the primary ideological characteristic is so vastly different. The right have some good ideas, so does the left. We take the best of them. Out centralism the centralists.
It's not a different team, it's a completely different sport.
Because of the centers attack on 'leftism' that's only going to hot up, I think this would be a good way to parry the blows. Instead of arguing that they're wrong about leftism, say yeah, we're not leftists tho.
At the very least it'll frustrate the argument.
Thoughts?
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I don't think it's just a matter of people not knowing about it. Through examination you can always derive the same structural elements that make up communism, and it's less important to describe a complex system that will satisfy certain criteria than it is to make sure that criteria is requisite in how people discuss politics so they can come to those conclusions them selves, or at least be more receptive to them.
This is just what I've had to work with in my experience. I don't think people around me have ever had a particular issue with communism, or even much negative association with it, but more it lacks any meaning beyond being diametrically opposed to a capitalist system. What I fear to present is the idea that there is an entire catalogue of socioeconomic systems to choose from and Capitalism and Communism are just two of them. Capitalism being one that's good enough so lets stick with it.
Rather Communism needs to be a natural choice and Capitalism needs to be exposed as an artificial environment sustained through force.