1. 1.Twists applied immediately on the image data as it comes from the sensor. In this case, if you have two images of the same scene that differ in exposure, then the displayed images will differ in tint, and not just in luminosity, even if you use the Camera Raw or Lightroom exposure controls to adjust the images to the same exposure. So, for example, the same sweater (or whatever) in the two images would show as the center of patch 2 in one case, and the center of patch three in the other case.The reason is that the extent of the “twist” of any given pixel is being set by the original exposure, which was different. the up side of this situation though is that tint won’t change as you adjust exposure - it’s fixed.
2. 2.Twists applied after basic exposure adjustments. In this case, Camera Raw or Lightroom first apply basic exposure adjustments to the image - exposure, recovery, etc. Then the twists are applied. In this case, if you had the images of the same scene with different exposures, and adjusted them to the same, the tints would also match - so, in both cases the sweater would be the color of the inside of patch 2, for example. Note that prior to adjustment, the colors would be different - as you adjusted the exposure, the color in the image you were adjusting would move towards the color in the other image, and be the same once the exposure was the same. But that’s also the downside of applying twists after the exposure controls - colors change when you make exposure adjustments. Again, in some situations, that’s not great. Adobe’s default profiles (Adobe Standard, etc) work in this way.

1. 1.Make a profile Invariate. Making a  profile ivariate shifts any hue twists to before the basic exposure adjustments stage, so they are applied immediately. So an invariate profile won’t cause changes tint when you make adjustments to exposure settings. this is what you want to do the standard Adobe profiles if you want to keep the twists, but just prevent colors from changing when you adjust Carmera Raw or Lightroom’s exposure controls.
2. 2.Untwist a profile entirely. Untwisting a profile removes all hue twists completely. An untwisted profile will always keep colors true, regardless of original in-camera exposure, or changes to exposure controls in Camera Raw or Lightroom.
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4. Note: Generally, you will get better results from untwisting a profile than from making it invariate. Hue twists that are designed to be used post exposure adjustments often do not translate well into being used prior to exposure adjustments. Generally, I recommend that you try the untwisted version of a profile as a first step.

1. 1.Make a profile Invariate. Making a  profile ivariate moves any LookTable to the HueSatDelta table. Specifically, if a LookTable exists, and no HueSatDelta tables exist, dcpTool simply copies the LookTable to the HueSatDelta tables, and the removes the original LookTable. If both a LookTable and HueSatDelta tables exist, the tables are mathematically combined, and the combination becomes the new HueSatDelta table. Again, the original LookTable is then removed. If no LookTable exists, then the profile is already invariate.
2. 2.Untwist a profile entirely. Untwisting a profile removes all intensity (V in HSV) dependencies in both the LookTable and any HueSatDelta tables. Effectively, the twist at a single point in the twist table is chosen, and applied to all intensities. The twist chosen is that corresponding to a skin tone that is accurately exposed, thus preserving the profile's skin tone.