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This paper studies the problem of learning semantic segmentation from image-level supervision only. Current popular solutions leverage object localization maps from classifiers as supervision signals, and struggle to make the localization maps capture more complete object content. Rather than previous efforts that primarily focus on intra-image information, we address the value of cross-image semantic relations for comprehensive object pattern mining. To achieve this, two neural co-attentions are incorporated into the classifier to complimentarily capture cross-image semantic similarities and differences. In particular, given a pair of training images, one co-attention enforces the classifier to recognize the common semantics from co-attentive objects, while the other one, called contrastive co-attention, drives the classifier to identify the unshared semantics from the rest, uncommon objects. This helps the classifier discover more object patterns and better ground semantics in image regions. In addition to boosting object pattern learning, the co-attention can leverage context from other related images to improve localization map inference, hence eventually benefiting semantic segmentation learning. More essentially, our algorithm provides a unified framework that handles well different WSSS settings, i.e., learning WSSS with (1) precise image-level supervision only, (2) extra simple single-label data, and (3) extra noisy web data. It sets new state-of-the-arts on all these settings, demonstrating well its efficacy and generalizability. Moreover, our approach ranked 1st place in the Weakly-Supervised Semantic Segmentation Track of CVPR2020 Learning from Imperfect Data Challenge.