Every band I start to list has either done something rather disgraceful at a later date (like the Beatles playing behind that dismal Lennon demo
"Real Love" and calling it a new song, or the last few decades of the Rolling Stones, or Echo and the Bunnymen since 1986 on) or just been a shadow of themselves for so long that any question of relevance is shot (Motorhead, The Jesus & Mary Chain).
A couple have come back from their transgressions (Gang of Four, Wire) but I still don't feel comfortable with listing them as exceptionally relevant even though I'd go see them in a heartbeat.
One Brit band that is coming to mind who broke up before they ever sucked and who's reunion seems rather credible is Throbbing Gristle. So, I guess they're on my list. And I suppose My Bloody Valentine if the constant threat of reunions is overlooked (or if by some miracle it happens and they are surprisingly good). And I have yet to not love any records by Autechre.
So, I'm looking at a standpoint of conintued relevance. If I were to list historical relevance, the list would be so long as to be ridiculous.
I can think of a handful of American groups that meet that continued relevance criteria (Mission of Burma, Rocket from the Tombs...), and Sonic Youth who pulled it off without even breaking up.