Originally Posted by genec
Everything I have ever seen has the bar tape wrap starting from the ends of the handlebars and working toward the center, where the tape is then secured with electrical tape or something similar.
Look closely at the tape as it's currently on the bar. One (longitudinal) edge is "free", the other is covered by the next wrap.
Now think of the direction of the pressure applied with your hands resting on the outer portion of the bars, just above the brake hoods. Wrapped from the bar center towards the ends, the force from your hands is applied against the "free" tape edge, tending over time to roll it back into an unsightly mess. End --> center wrapping prevents this: the direction of your hand-friction is "with" the wrap.
More graphic example: imagine your house shingled with each shingle overlapped on the bottom rather than the top. Force applied in the form of rainwater causes more than cosmetic problems!