I've just gotta chime in here and say how much I used to be obsessed with The Blood Brothers. Watching their DVDs now, or if you saw them live when they were together, was an experience to itself. Not only did their songs defy convention, but their method of playing shows was almost awkward with their mannerisms and musical style. I've also listened to and seen HORSE the band play back when I went to UMD, though I think they came after BBs.
Innovative? How about the grand-daddy of all pseudo emo-screamopunk: Rites of Spring. Totally unheard of to take that punk raw nature and actually put feeling behind it, but also innovative in their song writing. Or how about Husker Du? Zen Arcade still stands the test of time as an incredibly innovative album in both concept and execution. The Dreams - Reoccuring songs still baffle me.
One last band to bring up, and one of my two favorites: Planes Mistaken for Stars. Starting off as Screamo-Punk and then becoming god-knows-what at the end (I think that Mercy is one of the weaker albums, but as usual with them, they defied some major musical logic even there). The two most innovative things I can say about them are their mixing style, which places the bass in a very weird place in the soundscape as well as how the guitars are written and performed with deliberate noise and mis-hits, and Garret's vocals, which during the bulk of their existence consisted of half-swallowing the words while screaming on-key. The lyrics were also extremely cryptic but poetic, with that particular vocal tone everything came through as this beautiful noise that I just couldn't get enough of. I would highly recommend the albums Fuck with Fire and Up in them Guts to highlight their innovation. Maybe I'm just a big fanboy of theirs, but I've heard things from them that I just can't place anywhere else. INNOVATION!
Edit: Also, another amazing band full of innovation and just all-around FUCKEDUPNESS is Secret Chiefs 3. Words can't describe it at all, but I'll try: Hardcore Arabic Surf Rock Techno Punk Orchestral. Yeah, it's fourteen million times more bizarre and incredible to listen to than my words can convey, and absolutely innovative. Check them out for sure.