It is usual to consider data protection and learnability as conflicting objectives. This is not always the case: we show how to jointly control inference --- seen as the attack --- and learnability by a noise-free process that mixes training examples, the Crossover Process (cp). One key point is that the cp~is typically able to alter joint distributions without touching on marginals, nor altering the sufficient statistic for the class. In other words, it saves (and sometimes improves) generalization for supervised learning, but can alter the relationship between covariates --- and therefore fool measures of nonlinear independence and causal inference into misleading ad-hoc conclusions. For example, a cp~can increase / decrease odds ratios, bring fairness or break fairness, tamper with disparate impact, strengthen, weaken or reverse causal directions, change observed statistical measures of dependence. For each of these, we quantify changes brought by a cp, as well as its statistical impact on generalization abilities via a new complexity measure that we call the Rademacher cp~complexity. Experiments on a dozen readily available domains validate the theory.