Your mechanic is nuts. First, the cylinder does not get touched. The head comes off, there are 2 pieces to that (a cam cover on top of the head). Any reputable automotive machine shop can get new valves, how many depends on the condition of the other 19 but usually just the exhaust valves, new valve seats, and valve seals for all 20 ($2 each).
If you went to a Volvo dealer, they would want to replace the entire engine with a reconditioned engine for around $5,500. And that is outrageous!
I have to question how they determined that a valve broke. Just checking compression isn't good enough. They would have to lower a scope into the spark plug hole, and take a picture.
Your car is in the wrong place.
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My back feels better when I sit in a Volvo seat
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