Previous approaches for computing duplicate-sensitive aggregates in wireless sensor networks have used a tree topology, in order to conserve energy and to avoid double-counting sensor readings. However, a tree topology is not robust against node and communication failures, which are common in sensor networks. In this paper, we present synopsis diffusion, a general framework for achieving significantly more accurate and reliable answers by combining energy-efficient multi-path routing schemes with techniques that avoid double-counting. Synopsis diffusion avoids doublecounting through the use of order- and duplicate-insensitive (ODI) synopses that compactly summarize intermediate results during in-network aggregation. We provide a surprisingly simple test that makes it easy to check the correctness of an ODI synopsis. We show that the properties of ODI synopses and synopsis diffusion create implicit acknowledgments of packet delivery. Such acknowledgments enable energy-efficient adaptation of message routes to dynamic message loss conditions, even in the presence of asymmetric links. Finally, we illustrate using extensive simulations the significant robustness, accuracy, and energy-efficiency improvements of synopsis diffusion over previous approaches.