Cross-impact, namely the fact that on average buy (sell) trades on a financial instrument induce positive (negative) price changes in other correlated assets, can be measured from abundant, although noisy, market data. In this paper we propose a principled approach that allows to perform model selection for cross-impact models, showing that symmetries and consistency requirements are particularly effective in reducing the universe of possible models to a much smaller set of viable candidates, thus mitigating the effect of noise on the properties of the inferred model. We review the empirical performance of a large number of cross-impact models, comparing their strengths and weaknesses on a number of asset classes (futures, stocks, calendar spreads). Besides showing which models perform better, we argue that in presence of comparable statistical performance, which is often the case in a noisy world, it is relevant to favor models that provide ex-ante theoretical guarantees on their behavior in limit cases. From this perspective, we advocate that the empirical validation of universal properties (symmetries, invariances) should be regarded as holding a much deeper epistemological value than any measure of statistical performance on specific model instances.