I can't guarantee this won't break something on your car, but at least on my visors, the attachment arms just wiggle and pull out of the visor. I assume that is what you have already done when you said you looked inside.
Attached is a picture of one of the attaching arms. Notice there is a part of the rod that looks like it has a rectangular area with some black smudges, that is a flattened area of the arm. I'm guessing that is what contacts something like a lever that touches that flat spot to provide the little click when it is in each "clicked" position.

Do both the mounting arms look like this, or is one possibly broken off ?

Notice there also is a tiny tab sticking out of the arm almost at the tip. I'm not sure what that piece might be for, but maybe that also contacts something to hold it in place.
Maybe you can look at both of the attaching arms if you pull them out and see if the tab sticks out at the end, or if that is broken off.
If one has a tab, and the other doesn't, you can try reversing the arms and seeing if you can get each side to click into a locked position. Even though the arms are mirror-images, you can try them in either side to see what the symptom is.
Both sides should provide the clicking action.

If those little tabs are broken off the tip, that might be the problem. Or if the arm broke off, and half of it is stuck inside the visor. Otherwise whatever is broken would be inside the visor mechanism. Maybe you will be lucky and it's just the mounting arm that's broken.
Let me know if that's the case, that one or both of the arms are broken.