Neighborhood enumeration is a fundamental concept in the design of metaheuristics. It is often the only principle of intensification present in a metaheuristic and serves as the basis for various metaheuristics. Given its importance, it is surprising that academic reporting on enumeration strategies lacks the necessary information to enable reproducible algorithms. One aspect of neighborhood enumeration in particular has been under the radar of researchers: the order in which neighbors are enumerated. In this paper, we introduce a versatile formalism for neighborhoods which makes explicit enumeration order and we analyse the impact of enumeration order on the outcome of search procedures with a small set of benchmark problems.