In a 2002 paper that is quite cited, Layman and Carsey conclude that
In recent decades, Democratic and Republican party elites have grown increasingly polarized on all three of the major domestic policy agendas: social welfare, racial, and cultural issues. [...] Mass attitudes toward the three agendas have remained distinct, but the parties in the electorate have grown more polarized on all three.
And they dubbed this process "conflict extension"
in which the mass response to recent elite-level developments should be limited to party identifiers, particularly strong partisans, who are aware of party polarization on each separate issue agenda.
I'm not going to go over their full argumentation here, but they found for instance that:
Importantly, there is no evidence of growth in attitudinal constraint among pure independents over time. The average loading of all issues on a single factor was lower in 2000 than it was in 1988, and the loadings of cultural issues display no pattern of growth. In fact, the correlation between the cultural factor and the social welfare and racial factors in a three-factor solution is statistically insignificant in each year.
There's a 2004 paper (freely available as full text) that more or less confirms that finding and puts it in less elaborate language:
Using NES data from 1972 to 2004, the authors model trends in issue partisanship—the correlation of issue attitudes with party identification—and issue alignment—the correlation between pairs of issues—and find a substantive increase in issue partisanship, but little evidence of issue alignment. [...] Levels of constraint vary across population subgroups: strong partisans and wealthier and politically sophisticated voters have grown more coherent in their beliefs.
Does this essential observation hold true of the US public (roughly) nowadays? Is issue alignment still largely limited to the party elites and "strong partisans"?