I tried this once, just to be obnoxious. Up front, let me say that it didn't make any difference and I didn't do it again.

Of course, I have a preferred point of view on which way to tape each section. I was primarily trying to keep the edge of the tape from rolling over because it was exposed in the direction I tended to slide my hands, rather than concealed. That meant wrapping from the middle of the bars downwards and from the middle upwards. You can do this another way using the same technique, though.

From half way up the bars, start wrapping downards. Start the wrap at an angle so you don't have a lot of overlap in the first wrap or so, otherwise you end up with a big bump there. Take it to the bottom and use your bar plugs to capture the ends. But here's the trick: After you've done just the first two wraps, insert the tag end of your upward-bound piece under the downward-bound tape. It needs to be inserted at the correct angle so it just starts winding right, and it needs to be inserted for a good full turn so it doesn't try to pull out. A little rim cement or rubber cement really helps hold it in place. Now you have the upward-bound tape anchored and ready to head up If you start it so the extra bulk is on the back of the bar, nobody will notice it. Take it to the top and tape it off.

If you really think you have a problem with unwinding, it's more a matter of whether the tape is wound with a left-hand screw or a right-hand screw (no jokes, please). Properly tensioned, it shouldn't be an issue, but if it is, this is your opportunity to reverse the direction and make sure it works fine.

If you prefer to start from the ends and work towards the middle (everyone here seems to have a different point of view), I did try this once and got some colored heat-shrinkable tubing in the electrical department of a Home Depot. Just get one of the really big sizes, just large enough to slip over the bars and bar tape, and then use a hair dryer to shrink it into place. It's extremely durable, the edges don't roll up or wear, and it actually protects the sides of your bar tape a lot when you lean the bike against a wall. You can use a couple different colors on top of each other, or match your bar tape, or whatever. It's pretty neat stuff.