Where do class and status fit in Jeffrey Alexander's Civil Sphere? They are not a separate sphere of their own. They are not simply part of the economic sphere. They matter a great deal in people's lives - they can't be ignored. He is not a Marxist, so he can't use classes as the basic building blocks of the social structure. Like all Durkheimians, he has a tough time translating the individual differentials created by the division of labor into the social groups - the layered, hierarchical social groups -- of the social classes.
I think Weber can be helpful here, as he usually is. Weber, against Marx, says that classes, defined as those who share an income level, are not really social groups at all except under rare circumstances. Status groups, on the other hand, are real social groups. They share a conception of how to live. They are measured at the same level of honor.
The civil sphere is where campaigns about the basic values of society are conducted. The basic values are coded, Alexander says, in binaries that underlie the motives, relations, and institutions of society. The normal form of a value competition is to argue that our side embodies the good side of the binary, while the other side is polluted by the bad side. And vice-versa. The outcome of the competition is decided when a majority, or perhaps a strategic minority (Alexander is not clear on this) accepts one side's characterization of good vs. bad, and rejects the other.
If one group in society is regularly and routinely thought by most people to embody the good side of the binaries, and another group is thought to embody the bad -- with many groups in the gradations in between -- then society would have a status hierarchy. This is a social structure. And in Alexander's theory, the natural place - maybe the only place -- this status structure could be located is in the civil sphere.
Alexander has resisted describing any social structures in the civil sphere. Structure, and especially hierarchical structures, are the uncivil tools of the uncivil spheres. Yet if there is a status structure in society (and it would be hard to find a complex society without one), and if the status structure is related to the actual values of society, then the civil sphere must be built on a hierarchy of status.