On the flip side, I'd argue that the same aspects of persona that people hate also attract a larger number of vocal supporters than are deserved. Strip away all the HARD HOCKEY MAN bravado and press conference moments and he's just another legacy coach who displays the same flaws in every gig he gets and hasn't accomplished much other than moral victories and temporary highs over the last 20 years.
Like, imagine if he was just kind of a boring, mild-mannered NHL coach with the same issues and outcomes. His teams consistently feature the worst PPs in the league; they play an unsustainable style based around chasing and shot blocking; often overperform early in seasons and collapse late; he makes a lot of really detrimental lineup decisions and manages players in odd ways. People mocked Alain Vigneault's obsession with Tanner Glass back in the day...Tortorella willingly played Nic Deslauriers in 80 games last season.
Any time a Torts thread comes up a whole bunch of fans of teams he's never coached pop up with "Love Torts, he's just what my team needs!" How many fans you see in the places he once coached going "Damn, I wish they never fired Torts"? Not too many, because those people have gone through the day-to-day grind of following his teams. It's no more magical or gratifying than a Hitchcock or Vigneault experience.
So the strong feelings in both directions seem to stem from how he holds himself publicly. Some people look at it and think there's something real there (ACCOUNTABILITY AND GRIND!) others look at it and see a lot of empty ego with repetitive results. It's like a politician or pop star dichotomy moreso than a sports figure. Problem is, he's part of a team--so when that persona wears out on the people in his locker room, it becomes just another flaw in his actual coaching work.