Like in the middle of this thing: https://www.etsy.com/listing/122696999/mid-century-cardboard-metal-arrow-game
I’m not talking about the spinning arrow, but the grommet-like circle in the center.
I just want to fasten two round cardboard pieces on top of each other, but I want to make it a permanent thing (so not the gold spike thing that splits into two after you poke it through the hole). Is it just a regular grommet? If not, what are they called and where might I find one…?
Spinner, I think. At least that’s what I recall them labeled in old games that included them.
You need a paper rivet.
that works fine if you put a regular hardware washer on the back of the paper, and then bend the spikes after they go thru the washer. Might work well to use a washer on both sides.
Years ago I worked for a small game company, one of my jobs was making spinners just like that. We used eyelets that are available at most craft shops to hold the spinner to the paperboard. I kept that stuff when the company folded and still have it in a box in the garage.
Chicago screws might work also
The ones you can get at scrapbooking stores/websites are usually called Eyelets. There are tools for setting them in place neatly, as well.
The setter usually looks like pliers or a belt hole punch. Another design is like a nail you place over the eyelet and hammer; this turns the end of the eyelet from a cylinder to a mushroom.
Thank you all 
I couldn’t find any paper rivets, but eyelets are everywhere. I’ll go to the craft store later and pick some up.
Paper Fasteners.
How would guards at Metropolitan Correctional Center in Chicago help?