Here is Betancourt’s essay on the topic, esp Section 3.2 which compares centered and non-centered. It is as @rpgoldman says, Betancourt argues that the “well-informed” parameters are harder to sample in the non-centered parameterization.
In an example lower down in the essay he uses a heuristic threshold of 25 observations, below which he assigns parameters to the non-centered parameterization (and above, to the centered). Of course, the specific threshold would depend on the data, the likelihood, etc, but it seems like “mixed parameterizations” can be helpful.