My understanding with PZI is you shoot through it when it's a false alarm (#s not too low, and drop not too steep). In those cases you just ignore it, continue with the dose you are on, and raise it if needed on whatever schedule you are following (i.e. don't raise the dose when you see the reds, but don't lower it either, and if you don't get improvement on that dose in a few cycles, then you go ahead and raise).
If it's a real alarm though, like actual low #s or really steep drops, then in my mind that is Somogyi rebound rather than liver training, and then you lower the dose, going especially gently with the shot right after the rebound so you don't risk another plunge in #s while the rebound itself is wearing off and the #s coming down on their own.
You wouldn't want to shoot through Somogyi IMO, at least on PZI, b/c that could actually cause a real hypo. I suppose you could shoot higher to counteract the red #s, but that seems pretty risky to me, you would have to be really on top of it to gauge when the reds are wearing off and then you really would need to lower the dose to avoid hypo & more Somogyi rebound. Unless you are dealing with ketones, to my mind it's better to ride out the reds and let the rebound clear itself up than to risk mis-timing things. Maybe a "for experts only" technique, I wouldn't try it myself unless there was a really compelling reason to do so.
I have seen panicky liver that looks just like Somogyi but without any low #s - for instance a drop that is on the steeper side but not really that steep, and then you see rebound, or sometimes you see it as PS that go higher as the nadirs are starting to improve. Sometimes kitties prone to that have done better on different insulins (I'm ready for the slap today! :lol: ), or shooting TID or changing the feeding schedule or something to soften the curves. Some cats seem more sensitive to the drops than others.
What I have seen a few times now, and I guess this is liver training school, is that if they get a few hours in blue #s they aren't used to, then you will often see a sky-high +12 (next PS), and sometimes the next cycle will be straight reds even on the same dose. But you can see from the #s there wasn't any steep drop or dangerous lows, so you know it's liver training and to be ignored, vs. Somogyi rebound & to be respected. In those cases I don't think there's any choice but to shoot through it - from what I have seen, the quickest route to more good #s is to force them to stay in good #s as long as you can til that liver settles down.