Large Language Models (LLMs) can achieve strong performance on everyday coding tasks, but they can fail on complex tasks that require non-trivial reasoning about program semantics. Finding training examples to teach LLMs to solve these tasks can be challenging. In this work, we explore a method to synthetically generate code reasoning training data based on a semantic inequivalence game SInQ: a generator agent creates program variants that are semantically distinct, derived from a dataset of real-world programming tasks, while an evaluator agent has to identify input examples that cause the original programs and the generated variants to diverge in their behaviour, with the agents training each other semi-adversarially. We prove that this setup enables theoretically unlimited improvement through self-play in the limit of infinite computational resources. We evaluated our approach on multiple code generation and understanding benchmarks, including cross-language vulnerability detection (Lu et al., 2021), where our method improves vulnerability detection in C/C++ code despite being trained exclusively on Python code, and the challenging Python builtin identifier swap benchmark (Miceli-Barone et al., 2023), showing that whereas modern LLMs still struggle with this benchmark, our approach yields substantial improvements. We release the code needed to replicate the experiments, as well as the generated synthetic data, which can be used to fine-tune LLMs.