Large language models (LLMs) have great potential for synthetic data generation. This work shows that useful data can be synthetically generated even for tasks that cannot be solved directly by LLMs: for problems with structured outputs, it is possible to prompt an LLM to perform the task in the reverse direction, by generating plausible input text for a target output structure. Leveraging this asymmetry in task difficulty makes it possible to produce large-scale, high-quality data for complex tasks. We demonstrate the effectiveness of this approach on closed information extraction, where collecting ground-truth data is challenging, and no satisfactory dataset exists to date. We synthetically generate a dataset of 1.8M data points, establish its superior quality compared to existing datasets in a human evaluation, and use it to finetune small models (220M and 770M parameters), termed SynthIE, that outperform the prior state of the art (with equal model size) by a substantial margin of 57 absolute points in micro-F1 and 79 points in macroF1. Code, data, and models are available at https://github.com/epfl-dlab/SynthIE.