The twisting is real in a sense, but it's more subtle than your picture suggests. The other answers, on the other hand, are correct in a technical sense but, I think, missing the big picture.
Imagine that you set up very far from the black hole and start sending out probes toward the black hole at regular intervals (as measured on your local clock). You hope to use these to mark out a coordinate system between you and the black hole. Each probe is left to coast after the initial impulse that sends it on its way, and each gets the same initial velocity starting at the same distance from your station (both, again, measured in your frame). While they remain far from the black hole, the spacetime is essentially flat, and they behave as you expect - staying on line with the black hole and equally spaced (in both coordinate and proper distance). Let's say you send out 1 per hour at 1 mile per hour. Then if you check each hour, your system extends one mile closer to the black hole, but you have one probe at the 1 mile mark, one at the 2 mile mark, etc. Still as measured by you in your reference frame.
Once they get close enough to the black hole, however, the black hole's effects will be observable on the probes. In the case of a Kerr black hole, they will start to orbit and separate in distance. If you keep sending out a stream of these hoping that they will be reference points on your grid, this twisting and stretching will be observable to you at your distanced position. This is a consequence of the fact that there is no globally inertial frame of reference in this spacetime. (Of course, in their own respective comoving frames, they are each still coasting inertially.)
You could try to overcome this by putting thrusters on your probes that fire in just the right way to "cancel" the effects of the spacetime curvature so that you get the regular grid that you wanted, as viewed by you at a distance. You can even hold them at a fixed coordinate distance if you have enough fuel and strong enough thrusters so that they appear stationary - That is until you try to put one in the ergosphere where no amount of thrust will be enough to avoid them appearing to you to orbit or inside the event horizon, where no amount of thrust can prevent their inward fall. Except in these special regions, you now have a situation where you view the probes to be at fixed coordinate positions in your frame. (But, correspondingly, each individual probe has to expend energy and is accelerating relative to a momentarily comoving reference frame.)
This coordinate system marked by the probes with thrusters is adapted to the black hole spacetime in way analogous to taking a corotating frame to study a planetary orbit. The thrust that you need to have each probe provide is equal and opposite to the psuedo forces associated with the non-inertial nature of the frame.
The other answers are pointing to special coordinates - along the lines of the second case - where any "twisting" is not evident in the form of the metric. Not all spacetimes have that, so it is something remarkable about Kerr and a few others. For theoretical coordinates, unlike our hypothetical physical probes that had size and mass, you can extend the system further toward the black hole. We can mark the mathematical coordinates however we want even if no physical item would be capable of station-keeping a constant-coordinate position. The answers are also, correctly, pointing to the fact that for the free-falling probes of the first scenario, each subsequent probe experiences the same gravitational effects as the ones before it when it passes by a corresponding point in space. Equally, the probes with thrusters in the second scenario, once they achieve their station, do not need a time-varying thrust to keep station. (Though each probe in the sequence needs a different thrust than the ones adjacent to it.) In that sense that spacetime is not dynamical such that some sort of "twisting" has a cumulative effect on the curvature.
