the old atheism allies itself with "left" politics, the new atheism allies itself with "right" politics;

I'm not sure left/right is the way to view politics over time.

Many old atheists lived before those distinctions were even created in the late 18th century during the French Revolution. The "right," at the time was loyal to religion and therefore could never be supported by any atheist. By original definitions, all atheists were on the left.

Of course, meaning changes over time and I'm no linguistic prescriptivist. But it also means someone being "right" or "left" today doesn't necessarily imply sympathy to our modern use of those terms.