tokens: that in the inertial frame only 2l + 1 out of (2l + 1) 2 peaks dominate, consistent with the matrix remaining approximately diagonal even at substantial obliquities (see Fig. 9 ). Closer inspection of how the shape of this dominant sub-multiplet changes with obliquity reveals that while this may be asymmetric in general, it is still possible for this to appear nearly symmetric for some values of β and have a spacing close to the value expected of pure rotation (Fig. 12 , lower left-hand panel). However, the centroid of this sub-multiplet is offset compared to the pure rotational multiplet. At larger field strengths (see Fig. 14 ), there may be many peaks of significant amplitude in the inertial frame. Notably, due to the heavy mixing, it would be possible to find symmetric sub-multiplets among these (e.g. bottom row, middle panel). Figure 12. Splitting diagram for a p-dominated mode of Model C, with n = −16 and l = 1 (i.e. that used in Fig. 9 ), under the default rotation and field strength listed in Table 1 , for six different values of the obliquity angle. Short, thick, horizontal black lines joined by thin dotted lines illustrate the splitting process, from the unperturbed mode (leftmost line at ω 1 = 0) into a multiplet of 2l + 1 modes in the corotating frame (middle group), and then further into (2l + 1) 2 modes in the inertial frame (rightmost group). In the rightmost group, the various modes have been horizontally offset for clarity. The different grey-scale shades correspond to the absolute value of the associated a m coefficient, with black being 1 (maximum amplitude) and white being 0. For comparison, horizontal coloured dashed lines mark the position of the non-magnetic rotationally split multiplet in the inertial frame. Figure 13. As for Fig. 12 , but a mode with n = −28 and l = 2 (i.e. that used in Fig. 10 ). Figure 14. As for Fig. 12 , but with a field strength three times higher. Frequency splittings for all l = 1 modes of Models C and D are summarized in Fig. 15 , where points are coloured according to the associated value of | a m |. The increased m -mixing with larger obliquity and field strength can clearly be seen across all modes, both p- and g-dominated. The main difference between p- and g-dominated mixed modes appears to lie in the centroid frequency of the multiplet: this is located towards systematically higher values compared to the unperturbed value for g-dominated multiplets, a feature not present in the case of pure rotation (as seen in Figs 7 and 8 ), where the centroid offset is zero for all multiplets. This offset between p- and g-dominated multiplets increases towards lower frequencies, where magnetic effects become more significant. Figure 15. Inertial-frame frequency shifts versus unperturbed frequency, for all l = 1 modes of Models C (left) and D (right). Values of the field strength and obliquity are shown in the panel headers. Within each panel, the different colours correspond to the corotating-frame
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