> ...at more than one company, I've been given specific instructions not to provide it (by the recruiting team).

What were you told?

That's the other side of this issue, of course. Because there are some things it would be hard to say out of politeness ("You pissed off the boss" or "People thought you were insufferable") or fear of legal repercussion ("We would never say this out loud, but one of the people on the team was sure that a woman would mess up the team dynamics").

So what did the recruiting team tell you [not] to say?