There was a time when lower middle class meant blue collar workers, middle-middle was white collar, and upper middle was doctors, lawyers, and other professions. All of these contrasted with an underclass of people who did not have jobs they could live on and with people who were independently wealthy and did not work unless they chose to. And despite spanning a range of economic and cultural differences, the middle class really did include a majority of Americans who had broad political interests and values in common.
You can call that "working class" if you like -- since it consisted of people who had to and did work for their living -- but nobody 50 years ago would have understood what you meant. "Working class" at that time was considered a vaguely old-fashioned term that if used at all was taken to mean the same thing as lower middle class.
The economic pressures of the last 50 years have destroyed the old middle class. With the loss of factory jobs and unions, the former lower middle class has been virtually obliterated. The former middle-middle, rather than being comfortable and socially valued has become insecure and beleaguered. And the upper middle has been picked off, turned into wannabe rich folk who complain about the fact that they can't quite afford genuine mansions and the very best private schools.
If anything, the efforts of the right have been directed towards destroying that ideal of middle class solidarity -- largely by way of wedge issues ranging from the Vietnam War and racism to abortion and same-sex marriage. In addition, the hopes of Marxism were pinned in the first place on the idea that industrialization had created a new class of factory workers with enough power over the operation of the system to make their voice heard. And to the extent that was ever true -- though briefly, and never to the extent Marx might have hoped -- its moment is over. Workers today have no power at all, and that may be our greatest problem.