The only confirmed Special Paramecia so far is Katakuri. He was first introduced as a Logia and Oda changed his DF to SParamecia. So now he is classified as Paramecia and has Logia tendencies. He has been seen with holes in his body from Luffys Gatling and recovering, he was blown in half by Ichiji and recovered, and he can create more of his substance; all things a Logia can do.
Mocha is a sticky, stretchy substance that can always regain shape after being tampered with. Again, we see this from Ichijis explosive punch.
The Mochi Mochi no mi is a SParamecia because of the gift of shape manipulation. Mochi can be contorted without harm and this is a greater power than being Mochi itself. Katakuri has obtained pseudo immortality.
Luffys Haki punches passed right through Katakuri because that is the nature of Mochi itself. A Logia would have been effected because a Logia isn’t immortal and Haki bypasses the intangibility because it is like invisible armor(said by Rayleigh) and touching your opponents invisible armor lets your strikes get through.
Now for Katakuri, he never had intangibility and Haki isn’t needed to touch him so what’s happening with him is that even though he gets hit he will always be mochi and has a chance to recover from life threatening injuries. For instance, when Jozu hit Crocodile with Brilliant Punk he received damage. Katakuri would feel the attack and be split if he doesn’t use Haki because that is the nature of a mochi body. Proof that Katakuri has felt every attack he has been hit with so far is that he went out of his way to counter Red Hawk with a Haki fist. If Red Hawk connected it would have hurt him because of the fire. He also avoids Jinbe; before the Mirror World the only attacks that he felt the need to try and avoid were a fire punch and Fishman Karate attacks. The only attacks that seem like obvious weaknesses to mochi, fire and water. The only question about this is why would he take Luffys other punches if he felt them? That would be because his base stats dwarf Luffys, Doffy stood calm against G2 attacks so they are nothing for Katakuri.
Without a good Haki defense, Katakuri will always split and reshape after attacks because he is mochi. Blowing his neck from his body would only result in him reforming because of his DF. Giving him pseudo immortality.
Now on the other hand, if you have a Mochi weakness than he is at a disadvantage and he would have to use his Haki more. A Red Hawk would do a lot more damage to him than a regular bodied person. So he isn’t immortal to whatever affects mochi but as far as other attacks and even Haki, he will reform and would be hard to put down even by people stronger than him.
The definition of Special Paramecia: A DF that has a better secondary function as a result of its primary function.
Katakuri has pseudo immortality and it’s a side affect of the primary function of his fruit which is to turn his body into mochi and allow him to stretch and stick things together.
Katakuri splits from all attacks without his Haki but he just won’t give attention to attacks that he deems he shouldn’t(he’s arrogant).
*Theory by AnonymousStrawHat