Businesses might roll up their mats and stow them inside when they’re closed. I think the co-op student store I used to work for did that. I’ve never heard of people doing that at private homes though. We’re not really a drop-by for a visit culture anyway, whatever the mats might say. We tend to make advance plans before visiting. Maybe long ago or in other areas?
However, I have seen a mat with a cleverly devised scrawl that said Come In right side up, and Go Away when turned around. That’d get the message across.