Many existing approaches for unsupervised domain adaptation (UDA) focus on adapting under only data distribution shift and offer limited success under additional cross-domain label distribution shift. Recent work based on self-training using target pseudolabels has shown promise, but on challenging shifts pseudolabels may be highly unreliable and using them for self-training may cause error accumulation and domain misalignment. We propose Selective Entropy Optimization via Committee Consistency (SENTRY), a UDA algorithm that judges the reliability of a target instance based on its predictive consistency under a committee of random image transformations. Our algorithm then selectively minimizes predictive entropy to increase confidence on highly consistent target instances, while maximizing predictive entropy to reduce confidence on highly inconsistent ones. In combination with pseudolabel-based approximate target class balancing, our approach leads to significant improvements over the state-of-the-art on 27/31 domain shifts from standard UDA benchmarks as well as benchmarks designed to stress-test adaptation under label distribution shift. Our code is available at this link.