Engineered or hardwood, rubber backed rugs can be an issue. I, foolishly, put one of those cheap rug pads under an accent rug. Yikes! I had to use GooGone and a soft scrub brush to get up the residue.

Some of my pads cost more than the rugs themselves, but like I said, I'm not gonna risk that nightmare again.

Throw them out.

As another who had to scrape off latex residue, I vote to only use rugs with felt rug pads underneath to protect the floor. I never want to go through that nightmare again.

Rubber backed rugs belong in a bathroom or maybe a tile entry thats it . If you have to keep them then get the best underpad made for hardwood floors.

As you can see, the verdict is "no rubber backed" anything with hardwood. Just because you move them around doesn't mean they won't damage your floors. Rubber has petroleum based plasticizers that keep it soft and malleable. It's the plasticizers that cause the damage. The petroleum based stuff is very damaging to polyurethane (which can be petroleum based as well). And that's where things go sideways.

Even to 'solid state' products can and will have chemical reactions to one another. It just takes a bit more time to do it..that's all.

I would ASSUME the rug backing WILL damage your hardwoods. That's the safest way to look at this. Go ahead and go rug shopping. These other rugs are going to be used in other areas...where stone and tile have remained.