When sex characteristics develop in ways that do not conform to binary models, dilemmas arise regarding how to understand the situation and what terminology to use to describe it. While current medical nomenclature suggests that it should be understood as a disorder of sex development (DSD) prompting medical responses, many describe intersex as a human variation in sexed embodiment that should be protected under discrimination laws. These opposing perspectives suggest different principles to employ in responding to dilemmas about gender assignment, early genital surgery and full disclosure of medical information. In this discursive psychological study, we explore how lay people, without prior knowledge or experience of intersex/DSD, make sense of these dilemmas and the underpinning discourses giving rise to how they talk about these situations. By using the discursive framework of ideological dilemmas, we analyse how people make sense of sex and gender (as binary or non-binary), how they deal with difference (as problematic or not), and how they understand who is in a position to make decisions in such situations. We conclude that engaging with dilemmas in-depth is more constructive than favouring one principle over others in moving social science research, reflexive clinical practice, and wider political debates on intersex/DSD forward.