Thinning is a commonly used technique based on iterative object reduction to obtain centerlines of segmented binary objects. Parallel reductions delete multiple points from the picture at the same time. For parallel subiteration-based thinning algorithms, an iteration step is composed of some subiterations, each corresponding to a specific deletion direction. The results produced by such algorithms usually depend on the order of these directions. This work presents the very first order-independent 4-subiteration thinning algorithm, which generates the same centerlines for arbitrary permutations of the four deletion directions and preserves topology for all possible objects.