In the social sciences, people typically report and interpret an odds ratio (OR) < 1 for a binary predictor which has a negative association with a binary outcome.
According to this article, the arrangement of the predictor variable which causes an OR of < 1 "produces a result that can only be interpreted as the odds of the first group experiencing the event is less than the odds of the second group experiencing the event. The degree to which the first group’s odds are lower than that of the second group is not known."
They and others suggest that one should reverse the order of the reference and comparison group of the binary predictor to derive an interpretable OR.
To me, this makes sense. But what if you have a continuous predictor variable which is negatively associated with a binary outcome variable? If, like me, you only have logits/log odds - which in this case would be negative - would you exponentiate the absolute value of the logit to get an interpretable odds ratio (where exponentiating the negative logit would otherwise provide an OR < 1)? The result would thus reflect the OR for a one-unit decrease in the continuous predictor varibale.