Motivation: Convolutional neural networks (CNNs) have been tremendously successful in many contexts, particularly where training data is abundant and signal-to-noise ratios are large. However, when predicting noisily observed phenotypes from DNA sequence, each training instance is only weakly informative, and the amount of training data is often fundamentally limited, emphasizing the need for methods that make optimal use of training data and any structure inherent in the process. Results: Here we show how to combine equivariant networks, a general mathematical framework for handling exact symmetries in CNNs, with Bayesian dropout, a version of MC dropout suggested by a reinterpretation of dropout as a variational Bayesian approximation, to develop a model that exhibits exact reverse-complement symmetry and is more resistant to overtraining. We find that this model combines improved prediction consistency with better predictive accuracy compared to standard CNN implementations and state-of-art motif finders. We use our network to predict recombination hotspots from sequence, and identify binding motifs for the recombination-initiation protein PRDM9 previously unobserved in this data, which were recently validated by high-resolution assays. The network achieves a predictive accuracy comparable to that attainable by a direct assay of the H3K4me3 histone mark, a proxy for PRDM9 binding. Availability: https://github.com/luntergroup/EquivariantNetworks. Supplementary information: Supplementary data are available at Bioinformatics online.