Knowing how the major chondritic components evolved and what their initial compositions were is pivotal for our understanding of the processes that shaped the early Solar System. Here, we have extended to the CR chondrites our testing of chondrule-matrix complementarity and the four-component model, i.e., two very different explanations for the bulk compositions of the carbonaceous chondrites and their components. Combining point-counting with electron microprobe analyses, we have analyzed four relatively primitive Antarctic CRs and the fall Renazzo. Our results for the abundances of chondrules and matrix are in good agreement with literature data, and confirm that these abundances vary considerably amongst the CRs (80.4 ± 2.3 wt.% and 18.5 ± 2.8 wt.%, respectively, in the four Antarctic CRs vs. 62.3 ± 3.4 wt.% and 33.2 ± 2.2 wt.% in Renazzo). The significant differences make the determination of the average properties and bulk compositions of the CRs problematic. This is particularly true for the volatile elements that were predominantly accreted in matrix. Nevertheless, all major and many minor element concentrations reported in the literature for average bulk CRs are reproduced here to better than 10%. By comparing our results to conventionally determined bulk compositions, we were able to verify the accuracy of our approach and identify elements likely affected by alteration or analytical artifacts (e.g., Ti, K, Co). Two particular compositional details of the CR chondrites investigated are (a) the relatively high contents of Mn in the chondrules compared to CO chondrules, and (b) the depletion of S in the matrix, relative to CI. In terms of the major elements Mg, Al, Si and Ca, our data suggest that unaltered chondrules and matrix exhibited CI-like relative abundances, supporting previous conclusions for the CO chondrites. Where observed, deviations of element abundances in the matrix from CI (Na, Mg, S, Ca, Fe, Ni) can be explained in terms of alteration (parent body and terrestrial) and pre-accretionary loss of forsterite and, possibly, sulfides. Overall, our results are more consistent with the predictions of the four-component model than they are with chondrule-matrix complementarity.