Protein Direct Coupling Analysis (DCA), which predicts residue-residue contacts based on covarying positions within a multiple sequence alignment, has been remarkably effective. This suggests that there is more to learn from sequence correlations than is generally assumed, and calls for deeper investigations into DCA and perhaps into other types of correlations. Here we describe an approach that enables such investigations by measuring, as an estimated p-value, the statistical significance of the association between residue-residue covariance and structural interactions, either internal or homodimeric. Its application to thirty protein superfamilies confirms that direct coupling (DC) scores correlate with 3D pairwise contacts with very high significance. This method also permits quantitative assessment of the relative performance of alternative DCA methods, and of the degree to which they detect direct versus indirect couplings. We illustrate its use to assess, for a given protein, the biological relevance of alternative conformational states, to investigate the possible mechanistic implications of differences between these states, and to characterize subtle aspects of direct couplings. Our analysis indicates that direct pairwise correlations may be largely distinct from correlated patterns associated with functional specialization, and that the joint analysis of both types of correlations can yield greater power. Data, programs, and source code are freely available at http://evaldca.igs.umaryland.edu.