Some issues that have been settled by the scientific community, such as evolution, the effectiveness of vaccinations, and the role of CO2 emissions in climate change, continue to be rejected by segments of the public. This rejection is typically driven by people's worldviews, and to date most research has found that conservatives are uniformly more likely to reject scientific findings than liberals across a number of domains. We report a large (N > 1, 000) preregistered study that addresses two questions: First, can we find science denial on the left? Endorsement of pseudoscientific complementary and alternative medicines (CAM) has been anecdotally cited as being more consonant with liberals than conservatives. Against this claim, we found more support for CAM among conservatives than liberals. Second, we asked how liberals and conservatives resolve dilemmas in which an issue triggers two opposing facets of their worldviews. We probed attitudes on gender equality and the evolution of sex differences - two constructs that may create conflicts for liberals (who endorse evolution but also equality) and conservatives (who endorse gender differences but are sceptical of evolution). We find that many conservatives reject both gender equality and evolution of sex differences, and instead embrace "naturally occurring"gender differences. Many liberals, by contrast, reject evolved gender differences, as well as naturally occurring gender differences, while nonetheless strongly endorsing evolution.