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Visual perception entails solving a wide set of tasks, e.g., object detection, depth estimation, etc. The predictions made for multiple tasks from the same image are not independent, and therefore, are expected to be consistent. We propose a broadly applicable and fully computational method for augmenting learning with Cross-Task Consistency. The proposed formulation is based on inference-path invariance over a graph of arbitrary tasks. We observe that learning with cross-task consistency leads to more accurate predictions and better generalization to out-of-distribution inputs. This framework also leads to an informative unsupervised quantity, called Consistency Energy, based on measuring the intrinsic consistency of the system. Consistency Energy correlates well with the supervised error (r=0.67), thus it can be employed as an unsupervised confidence metric as well as for detection of out-of-distribution inputs (ROC-AUC=0.95). The evaluations are performed on multiple datasets, including Taskonomy, Replica, CocoDoom, and ApolloScape, and they benchmark cross-task consistency versus various baselines including conventional multi-task learning, cycle consistency, and analytical consistency.