Because I run 3 cars, I bring a lot of tires to the track - both new and used. I tried a few things to organize my tires, but it really wasn't doing the job - so today, I spent some time to create my own tire box using some card stock and cellophane wrap. I did the pattern on the computer and just scored the edges before I bent them. Because of the size of the box, I ended up using two 8.5 x 11 sheets. I'm sure this will work with heavier stock cardboard as well, if you draw it out using a ruler. You'll also get a one piece box instead of two. Note that the sides of the lid are the same height as the bottom, so it doesn't collapse when you stack it.
With the opening, you can easily see the size of the tire and/or what brand they are. This also helps if you run different size tires on each wheel as well, because each wheel is laid out for you.
If you have a cool way of organizing your tires, please share it with us!