The novel coronavirus pandemic continues to ravage communities across the US. Opinion surveys identified importance of political ideology in shaping perceptions of the pandemic and compliance with preventive measures. Here, we use social media data to study complexity of polarization. We analyze a large dataset of tweets related to the pandemic collected between January and May of 2020, and develop methods to classify the ideological alignment of users along the moderacy (hardline vs moderate), political (liberal vs conservative) and science (anti-science vs pro-science) dimensions. While polarization along the science and political dimensions are correlated, politically moderate users are more likely to be aligned with the pro-science views, and politically hardline users with anti-science views. Contrary to expectations, we do not find that polarization grows over time; instead, we see increasing activity by moderate pro-science users. We also show that anti-science conservatives tend to tweet from the Southern US, while anti-science moderates from the Western states. Our findings shed light on the multi-dimensional nature of polarization, and the feasibility of tracking polarized opinions about the pandemic across time and space through social media data.