The breakdown is similar to the one in the whole electorate. A Democrat who is as liberal as a Republican is conservative is more likely to consider themselves moderate, after realizing that liberal is a word with a negative connotation to the American public.
Or.... they realize that they aren't actually liberal at all? Or conservative either for that matter. Neither are terms that I would be particularly proud to call myself. Embarrassed, more likely.
The same goes for the word "Independent". "Independents" are not politically independent--they are weakly partisan Democrats who refuse to associate themselves with the hard-left Democratic Party of today.
Why the partisanship? Why can't Independents be "weakly partisan" Republicans who refuse to associate themselves with the hard-right Republican Party of today? Both are as bad as each other, and both are helping break this country clean in two.
Except that, according to most surveys, "Independents" are well to the left of most Republicans on social factors, and only slightly to the right of the Democrats. Economically, they favor Democrat positions more strongly--usually much strongly--than the Republicans'. As a whole, they even dislike the idea of cutting taxes. So, my assessment of "Independents" as lapsed Democrats is not mere partisanship.
A generation ago (say, 1980), "Independents" would have been just another segment of the Democratic coalition. It was only recently that 'Democrat' has become nearly as strong a pejorative as 'liberal' is, so those weakly partisan Democrats have naturally disavowed their connection to the Democratic Party and now call themselves "Independent"