It just occurred to me that this has less to do with must and have to and more to do with the scope of negation. I'm not claiming this as a new discovery. I've probably read or heard it before and simply forgotten, but it just struck me now and I don't think I'll forget again.
With must, negation comes after must, or is part of it in the case of mustn't. Periphrastic have to, though typically takes its negation to the left and requires do support. Here are two examples:
- You mustn't do that.
- You don't have to do that.
- You have to not do that.