The influence of soil structure on the volume change behavior of clay soils has been often studied with reference to intrinsic properties and normalization of reconstituted clays. However, to date, there has been relatively little research conducted on the normalizing behavior of naturally and artificially structured clays. In this paper, the yielding modes and normalization, by introducing the void index, are clarified for the one-dimensional compression of natural Finnish clays. Based on 24 samples with different cement contents and curing times, two novel approaches are proposed to normalize the post-yielding range of artificially structured clays. Laboratory results show that approach I, i.e., linear extrapolation of the post-yielding line in a log-log graph, can normalize well the compression behavior of artificially structured clays. However, the normalized curves of the void index vs. the effective stress in the semi-log plot deviate from Burland's intrinsic compression line (ICL) at very high consolidation stresses. To address this issue, a novel formulation of the ICL is deduced from laboratory data for artificially structured clays. Approach II, i.e., redefinition of the constants of compressibility to modify the void index, results in closer convergence of the normalized compression curves. The normalized curves by the modified void index agree well with the proposed intrinsic compression line for artificially structured clays (SICL). Finally, the generalization of Burland's ICL can redefine the intrinsic formulations corresponding to different levels of the yielding stress by modifying the void index.