While it is tempting to think of closed time-like curves (CTCs) around rotating bodies, such as a black hole, as being “caused” by the rotation of the source, Andréka et al. pointed out that the underlying physics are not as straightforward as this, since such CTCs are “counter-rotating”, i.e., the time orientation (the opening of the local light cones) of the CTCs is opposite to the direction in which the singularity or the ergosphere rotates. It was also suggested that this is a generic phenomenon that calls for a deeper intuitive physical understanding. In this short note, we point out—with Kerr–Taub–NUT as an example—that CTCs are counter-rotating with respect to the local angular velocity of the spacetime, not the global angular momentum, nor the angular velocity of the black hole horizon, which makes the physical interpretation of CTCs being “caused” by a rotating source even more problematic.