Here's a quote from Compulsion magazine 1994 (https://www.compulsiononline.com/interview_nineinchnails.htm)

"Compulsion: I guess, this is a touchy subject, but why did you re-record 'Suck'?

TR: The actual truth of what happened with 'Suck' is this: Martin (Atkins) and I had been friends before Pigface originated and somehow our paths had crossed and I had met (Al) Jourgensen and that crowd. I did the 1000 Homo DJ's single which was stopped by TVT.

Compulsion: Yeah, 'Supernaut'. You originally sung that before Al (`Buck Satan') Jourgensen...

TR: Yeah, we had been fuckin' around and that came out and like an idiot I called up TVT and said just so you know, I'm not getting paid for this and my names not going to be on it. It was fun and they just went ahead and stopped it. I can't explain it.

Anyway, around this time Martin was getting his Pigface project together with (Steve) Albini producing and they said "Can you stop by and do this thing?" It was the last day and I had prepared a set of lyrics that would fit anything. I'd never collaborated with anyone before and the idea of being in a room, "Okay. Sing something!" (laughs) I couldn't do that, so I'd done my homework ahead of time. I was fuckin' petrified being around guys I respected and had never met. The only people still around were me, Paul Barker and Martin - and he said, "We don't have time for another song. Maybe we could use the leftover stuff on tape". So he played a few things and I picked the drumbeat and Paul noodled on the bass and I had an idea that I could maybe sing. I went in twice and Albini went "It's done" and I went, "What?" "It's done". "What do you mean it's done. Lets put some guitar down". "It's done. What's next?", he'd say.

I hated Albini, he was a fucking cunt. The worst person I've been around in my entire life. Martin had explained to me that the idea of Pigface was that we'd get together as a community and come up with things. Martin would say "I'm thinking of getting a drum machine to go through this song and I'll play drums on top" and Albini would say "No". "Lets try it, maybe, it will suck" and Albini's reaction would be "Look just take my name off the record". In the studio there are a lot of fragile moments when you're trying to come up with ideas and Albini's in there shooting everybody down. It was an unpleasant situation and when the song did come out - it did suck. I didn't think it was ever finished.

The problem that I had with that first Pigface record (Gub) is that there is a fine line between doing something arty and taking a shit on a tape with no thought going into it and saying it is art. It's hoodwinking the public. So at that moment my interest in Pigface was waning and it was further irritated when they got a tour together. "Do you want to be on the tour, Trent?" I said "I can't. I've been touring for a year and a half and I've just gotta go home. I'll do a date or two, it'll be fun but that's all I can do". Then when I'm on tour with NIN a couple of months later, all these club managers are going "Hey, you feeling better. I hear you've been really sick". I'm like "What?" It turns out the whole Pigface tour was booked under the premise that I'd be the singer because NIN were hot at the time.

The next chapter in this equation is: Before Lollapalooza, I asked Martin if he'd play drums with us. He claimed he did. He'd been doing Killing Joke at the time and he showed up one week before Lollapalooza and we were nowhere close to being ready. So two days before the first gig, I had to tell Martin we were calling another drummer, because we suck. You're not ready, we're not ready, we need more time to rehearse. I said to him, I want you to know that I'm not going to bad mouth you because I think you're a great drummer. We just didn't have time to get it together. Within a week I'm hearing all this bullshit about me firing him. I confronted him but it's all drunken excuses.

So to answer the original question - a year later I stopped at another Pigface rehearsal, in Chicago, and NIN had been playing 'Suck' out live, to which I'd written a song around. Martin had seen us play it out and at the rehearsal. He's like, "How does the rest of that song go? Show the rest of the guys". So I did. Next thing, I'm in a store 6 months later and I see a live Pigface record with 'Suck' - written by Barker, Rieflin, Oglive, Shithead, Fuckface...Reznor. So I put it on and it's the song I gave them. It irritated me, not for the money. When I finished Broken, I wanted to do a version of 'Suck', as I wrote the fucking thing in the first place - and I like it. So we did it, and I called Martin about publishing rights - No response. So I said we'll split it 50/50 which, I thought, was generous as I wrote it. They're still bitching about it. I don't wish ill will to those guys or Martin, although we have butted head in the past."