yeah I know what you mean, it seems like we would want to check raise without a club in order to increase the likelihood that IP had a flush draw himself. however, if you use PIO somewhat often you will see that it prefers to develop more of a polarized turn betting range in most scenario's, and with low SPR situations it is pretty disastrous when we bet a hand such as JTcc and get jammed on. where as if we have JThh we can comfortably bet because if we get jammed on we can just snap fold.

so a hand such as JThh won't have much equity at all as a check back (mainly hoping to spike a pair vs an AK type hand and then x down) and we will have a bunch of sets/two pairs/straights/over pairs etc so we have tons of value hands so we of course want to develop some bluffs as well. JTcc on the other hand has a bunch of EV as a check back, it gets to realise all of it's equity and also bluff sometimes when x to and we brick. also, like I said, it suffers greatly from a x raise so is happy to check back and take a free card.

so assuming that we are checking back FD's more often than we are betting them on the turn, and that we are betting more of a polarized range with suited combinations that do not interact with the board (hearts and diamonds and one club/spade AK/AQ type hands) then JJ with a club does a good job of unblocking these combinations, and benefits from a bunch of protection by check raising. as well as folding out hands that may have bluffed us off of our hand on certain run outs but now have to fold.

Is it to potentially bluff flush completing rivers?

this is also a possibility. on a river card such as the A/Kc then it allows us to have some bluffs, although I think that the aforementioned points I mentioned were more of the reason that PIO would check raise with JJxc combinations as opposed to other ones. but this could certainly be a small factor/theory behind taking such a line.

good question imo :)

Ive heard good players say that "we do a good job of blocking villain's higher-equity hands" that continue vs a bet/raise. I've also heard that "we have more equity when the flush does come in". For example if the river came the 3c we would have a more profitable bluff-catch after XRing turn? Im not really sure if the context of your question matches the context of what im saying but i hope it can continue the discussion of something that is interesting if not important. I myself dont have a game plan for XR turn (and playing rivers when called) on this particular texture with this hand category but ive seen this trend in solver outputs in other spots and would love some context for this mystery.

ahh, blocking villains high equity continues on turn after getting x/r'd is a very good point

Also i remember the Tyler vid where he rails nosebleeds and cindy XR vs linus on Kh7c3h or something like that and barreled pot on an off suit T and Tyler said Linus prefers to continue ott w/ QhQx compared to QQ without a heart because when he rivers a Q it doesnt bring in the flush. That sounds like such a small effect because of how unlikely it is to happen but i guess when you consider how bloated the pot is becoming in both my example and yours it makes sense to block a possible "cooler".