We are able to compare the loudness of a tone to the brightness of a visual stimulus, and vice versa. This may be explained by the long-standing assumption of a common representation of perceived intensity that is shared by almost all modalities. Luce, Steingrimsson, and Narens (2010, Psychological Review, 117, 1247-1258) formalize this idea within a cross-modal version of the theory of global psychophysics, which can be empirically tested in a parameter-free way through the axiom of cross-modal commutativity of successive magnitude productions. The paper provides a theory-based analysis of data on this axiom collected by Ellermeier, Kattner, and Raum (2021, Attention, Perception, & Psychophysics, 83, 2955-2967), which is grounded on a recently suggested extension of the global psychophysical approach to cross-modal judgments (Heller, 2021, Psychological Review, 128, 509-524). This theory assumes that stimuli are judged against respondent-generated internal references which are modality-specific and potentially role-dependent (i.e., sensitive to whether they pertain to the standard or the variable stimulus in the performed cross-modal magnitude production task). The analysis reveals a massive and systematic role-dependence of internal references. This leads to predicting small but systematic deviations from cross-modal commutativity, which are in line with the observed data. In analogy to a term coined in the context of Weber's law this phenomenon is referred to as near-miss to cross-modal commutativity. The presented theory offers a psychological rationale explaining this phenomenon, and opens up an innovative approach to studying cross-modal perception.