Products disappear when changing category

We have products in a category but we want them to be in another just to minimize and make it easier to find.
However when changing category, the products disappears on the live site. They’re still in the backend, but I can’t make them visible on the site?!
Any thoughts on that?

Please provide a read only link so a member of the community can try to assist you

Ok. Hoped perhaps someone recognized the issue before.
It’s now done.

A bit more specific:

Previously I tagged a Ecommerce Product to a CMS collection called “product categories”. Now when I removed that tag, it is not displayed in my main CMS Collection.

It’s like the original tagging is saves somewhere. What can I do?

See demo here: