There's insurance that allows the mortgage company to recoup their losses, and then there's the insurance that allows you to rebuild. And then there's PMI, which is insurance the mortgage company can take out in case you default on your loans. We have no PMI, but the mortgage company (our credit union) automatically took out their own insurance (already factored into the loan) and require us to have our own.
I think that's how it's working out, anyway. The one the mortgage company took out is called "Hazard Insurance."