The short answer is that a car can become clean if it is rebuilt from salvage. The process can be lengthy, and the requirements vary from state to state. Generally speaking, whenever a salvage vehicle is rebuilt for operation, it must go through an inspection to qualify for road use. The DMV will title the car, but with the rebuilt/restored ‘brand’ on it, stating that it was rebuilt from salvage. Further certification and qualifications must be obtained and provided to the DMV in order to clear the title of the rebuilt brand. Check with your local DMV for specific requirements.
For more info, see A Rebuilt Car Title: Pros vs Cons