I am going to go against the general concensus here and say that if cleanliness is a concern, you may want to try an apron. I would absolutely not suggest that you hold the tire, or worse, a mounted wheel/tire assembly away from your body.
I don't do tire work as much as I used to, but I have done an obscene amount of tires at work in the past. If you are doing them what I would consider to be the right way, you are going to end up dirty. When you take a wheel off of a car, hang onto the wheel and pull it in close. If it is heavy, don't feel bad about supporting the wheel on your leg. Trust me, I am strong enough that I could hold a wheel away from my body, but I don't. Here's why.
There is only so much punishment that your back is able to take. Pulling a wheel into your body and supporting it on a leg if necessary takes a great deal of strain off of your back. Sure, you may get dirty, but dirt washes off. There have been a great many people who have washed out of a career in automotive repair because their bodies couldn't take the abuse any more. Earning a living slinging tires is hard, dirty work, and that is assuming you do everything the easy way. There is no need to make your job harder than it needs to be.
Now, I'm also not saying that every greasy looking guy is somehow a better tech than you because he managed to get covered in dirt today. All I'm saying is.... take care of your body, you only get one.
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