Can the university cut the notice period short? I am negotiating a position at another university. I’m on a permanent contract and a UK university that states six months notice period on both sides.

This is fine by me since I have many loose ends I need to take care before leaving, work and family wise.

My question is: If I give notice to the university, can they cut the notice short? E.g., can they ask me to leave earlier? Or, are they obliged to respect the six months, even though I was the one giving notice?

I don't believe they will do that, because they need me to stay until they recruit someone else. However, it would be good to know that once I ”pull the trigger”, I have a six months buffer to settle everything. For obvious reasons, I don't want to make this question to HR.

The university policy on notice periods is very concise. It says that unless otherwise agreed in writing, the notice period corresponding to my grade and position is six months from the university and me.
I suspect (but have no legal standing) that they are obligated to pay you through the notice period even if they ask you to leave early. 

But it also seems like they still are bound by a six month notice period rather than suddenly getting a shorter one because of your action. 

But suspect and seems are pretty dangerous to rely on from someone who is just speculating. And, no one here can give you legal advice even with full information. We can only guess and hope we are correct. But you are the one on the line here. 

I worry, also, that you have an adversarial attitude about HR. I suspect (that word again) that they are legally obligated to give you proper advice. They probably don't require all of the details of your situation to do so.