Why is track cycling done counter-clockwise?
The UK National Cycling Centre FAQ (PDF) says:

Why do the riders go anti clockwise?
The Chariots in Roman times raced this way round, and athletic races and most other sports have followed in the same direction.

The Straight Dope says:

How do these things get started? I've gotten several letters asking why races are &quot;always&quot; counterclockwise, and I notice my fellow toiler, Omni magazine game czar Scot Morris, has written a lengthy treatise on the subject. Scot came up with the following list of counterclockwise phenomena: the Indianapolis 500 and other auto races, track and field events, Roller Derby, indoor bicycle races, horse races, speed skating, merry-go-rounds and other carnival rides, revolving doors, the chariot race in Ben-Hur, the customary flow of people around an ice-skating rink, the usual direction in which people spin Hula Hoops, the base runners in baseball, cable-operated model airplanes, and tornadoes and hurricanes in the Northern Hemisphere. ...
On the theory that there might be some dark link to our primeval past, Morris asked several distinguished anthropologists, but no luck. He morosely concludes, &quot;the bias toward moving our whole bodies in counterclockwise cycles undoubtedly can be traced back to the right-handedness of our species and of every human society yet discovered, but how the one led to the other is unclear.&quot;

I didn't find it in the general UCI track rules, but it's in the track rules (PDF) of the UCI's World Cycling Centre in Aigle, Switzerland:


All riders must ride in an anticlockwise direction.