Higher-order mutation has the potential for improving major drawbacks of traditional first-order mutation, such as by simulating more realistic faults or improving test-optimization techniques. Despite interest in studying promising higher-order mutants, such mutants are difficult to find due to the exponential search space of mutation combinations. State-of-the-art approaches rely on genetic search, which is often incomplete and expensive due to its stochastic nature. First, we propose a novel way of finding a complete set of higher-order mutants by using variational execution, a technique that can, in many cases, explore large search spaces completely and often efficiently. Second, we use the identified complete set of higher-order mutants to study their characteristics. Finally, we use the identified characteristics to design and evaluate a new search strategy, independent of variational execution, that is highly effective at finding higher-order mutants even in large codebases.