Digital badges have long been assumed to possess motivational qualities that could encourage learners to engage with learning content. However, prior studies have found the effects of badges to be complex, differing by learner, type of badge, and potentially other factors. Qualitative reports suggest that individuals' perceptions of digital badges may play a role in moderating badges' effects: badges are only motivating when they are perceived as relevant and desirable. In the current study, we examine longitudinal episodes of data from 2,410 middle and high school users of a badged online programming curriculum to test whether there is evidence that learners' perceptions of badges' relevance predict changes in engagement over time; and whether that relationship is equitable with respect to age, sex, and ethnicity. We also investigate whether a reciprocal relationship may exist in which engagement predicts relative increases in learners' perceptions of badges as relevant to them. Learners' positive perceptions of badges' relevance predicted rank-order increases in engagement over time. Further, this relationship did not vary by age, sex, or minoritized racial/ethnic status. In addition, higher engagement also predicted upward shifts in perceived badge relevance. These results suggest that learners' subjective evaluations of digital badges are closely related to changes in their engagement with program activities, that badged engagement neither widens nor closes educational equity gaps, and that learners' regard for badges and engagement in program activities may be mutually reinforcing.