Dentist dispute in Kirchberg - What consumer rights do we have?

Dear Luxembourg,
I need your help please. Dentist in Kirchberg in one of clinics promised to do dental night guard. On the first appointment she took all the measures. On the next appointment when she tested night guard it didn't fit my mouth at all. She was insisting this is correct the way it is and if i want to fix that we need to send it back and i need to pay extra. I explained her if i go to the tailor for a shirt and he will take all the measurements and the shirt will not fit, i will obviously not pay extra for that. The tailor will need to fix his mistake.
So the situation is the dentists insist i owe her 650 eur while I am rejecting her service. I didn't take night guard and i didn't also pay the invoice. The question is what rights do I have in this case? What can happen if i reject to pay?
Thank you.

Edit: Thank you very much for all the replies and very useful advices. They called me from this clinic (in the comments I've mentioned which one (i don't want to publish on the front page)) saying that the owner would like to meet me. So I accepted this very cheerfully expecting that he will do better job and take the responsibility for what is happening (i will keep you updated). This post is more for awareness what can happen to people in Luxembourg even at the doctors/dentists that have very good online reviews. Once it happens you start to understand the profit oriented motive is much more important than your health. Ok it's just a night guard in my case. It could be something more serious. Still lots of money is involved here no matter how much CNS would reimburse. Would still be cheaper for dental clinic in the epicenter of Kirchberg to keep good reputation and sometimes swallow down their own mistakes and failures instead of fighting with the customer whose story ends up on the platform like Reddit, Google reviews etc...

Edit 2: We negotiated with owner of the clinic and I left without paying anything also receiving anything. He provided me other options but this was my decision. Respect to him that he took this responsibility on him. But it could be also other way around. Lesson learned: be careful.