I see this saying all the time and I think it's flawed. The second sentence I don't have a problem with, but to say that there's not also a left/right divide there, I think is to misunderstand the nature of the left/right divide.
See, the left prioritizes the well being of those on the bottom. They want to end the hierarchy. The right prioritizes the well being of those on top. They want to uphold the hierarchy. If you are on the bottom, or care for those on the bottom, you should be a leftist.