How I "fold" wire-bead bicycle tires:
How | Why
Credits:  Thanks to my daughter for "hand work".  I learned this method around 1979 in a Bike Warehouse catalog (Bike Warehouse has since become Bike Nashbar).

HOW:

First, pinch the tire into two loops.  It's better if one loop is larger than the other (unlike the photo).

Next, fold over one of the loops as shown:

The idea is to get the three "loops" that are shown above to be about the same size.  Next, "merge" the two loops shown above on the right:

Finally, bring the left loop up:

...and over:

Slide around a little so the three loops are equal-sized, and adjust the bead so there aren't any kinks in the sidewalls.  Tie loosely to keep it from springing open.

WHY:

(To take with you on a tour, for a spare, or to ship via UPS/USPS in a box)
Alternatives are to not fold at all, which is a bit unwieldy, or to fold in two.  I recommend against a two-loop fold, which can kink the bead as shown below: