While invariances naturally arise in almost any type of real-world data, no efficient and robust test exists for detecting them in observational data under arbitrarily given group actions. We tackle this problem by studying measures of invariance that can capture even negligible underlying patterns. Our first contribution is to show that, while detecting subtle asymmetries is computationally intractable, a randomized method can be used to robustly estimate closeness measures to invariance within constant factors. This provides a general framework for robust statistical tests of invariance. Despite the extensive and well-established literature, our methodology, to the best of our knowledge, is the first to provide statistical tests for general group invariances with finite-sample guarantees on Type II errors. In addition, we focus on kernel methods and propose deterministic algorithms for robust testing with respect to both finite and infinite groups, accompanied by a rigorous analysis of their convergence rates and sample complexity. Finally, we revisit the general framework in the specific case of kernel methods, showing that recent closeness measures to invariance, defined via group averaging, are provably robust, leading to powerful randomized algorithms.