Bit old thread but...
I agree: St.Pauli, Celtic and Livorno are strongly and clearly left-wing teams.
I wouldn't be so sure about Liverpool and I don't know anything about Hapoel Tel Aviv or Omonia Nicosia.
Barcelona is not left-wing; is a nationalist team. Catalonian nationalism is fueled by conservative upper-middle class (like the currently ruling party) and so is Barcelona. Believe me: not a trace of socialism or left-wing politics in their fanbase or inside the club.
In Spain, Rayo Vallecano got an incredibly strong far left-wing fanbase and the whole club is related to that (except the managers, of course). Sevilla, Celta and Cádiz got huge left-wing fanbases too, but they're not so dominant as in the rest of the teams we've talked about.