Much research shows academic self-concept and achievement are reciprocally related over time, based on traditional longitudinal data cross-lag-panel models (CLPM) supporting a reciprocal effects model (REM). However, recent research has challenged CLPM's appropriateness, arguing that CLPMs with random intercepts (RI-CLPMs) provide a more robust (within-person) perspective and better control for unmeasured covariates. However, there is much confusion in educational-psychology research concerning appropriate research questions and interpretations of RI-CLPMs and CLPMs. To clarify this confusion, we juxtapose CLPMs and RI-CLPMs relating math self-concept (MSCs), school grades, and achievement tests over the five years of compulsory secondary schooling (N = 3,425). We extend basic models to evaluate: directional ordering among three rather than only two constructs; longitudinal invariance over time (multiple school years) and multiple groups (school tracks); lag-2 paths between non-adjacent waves; and covariates (gender, primary-school math and verbal achievement). Across all basic and extended RI-CLPMs and CLPMs, there was consistent support for the REM bidirectional-ordering hypothesis that self-concept and achievement are each a cause and an effect of the other. Consistent with the logic of these models, extensions of the basic models had more effect on CLPMs, but the direction and statistical significance of cross-lagged paths were largely unaffected for both RI-CLPMs and CLPMs. This substantive-methodological synergy has important implications for theory, methodology, and policy/practice; we support the importance of MSC as a predictor of subsequent achievement and demonstrate a more robust methodological framework for evaluating longitudinal-panel models.