In recent years, large language models (LLMs) have demonstrated strong performance on multilingual tasks. Given its wide range of applications, cross-cultural understanding capability is a crucial competency. However, existing benchmarks for evaluating whether LLMs genuinely possess this capability suffer from three key limitations: a lack of contextual scenarios, insufficient cross-cultural concept mapping, and limited deep cultural reasoning capabilities. To address these gaps, we propose SAGE, a scenario-based benchmark built via cross-cultural core concept alignment and generative task design, to evaluate LLMs' cross-cultural understanding and reasoning. Grounded in cultural theory, we categorize cross-cultural capabilities into nine dimensions. Using this framework, we curated 210 core concepts and constructed 4530 test items across 15 specific real-world scenarios, organized under four broader categories of cross-cultural situations, following established item design principles. The SAGE dataset supports continuous expansion, and experiments confirm its transferability to other languages. It reveals model weaknesses across both dimensions and scenarios, exposing systematic limitations in cross-cultural reasoning. While progress has been made, LLMs are still some distance away from reaching a truly nuanced cross-cultural understanding. In compliance with the anonymity policy, we include data and code in the supplement materials. In future versions, we will make them publicly available online.