I don't know. I expect it to be symmetrical in that direction. Why should it behave different tilting left than right? It being periodic is much more unexpected to me. It makes sense but, without really thinking about it, I would have expected it to be more chaotic.

I dug up the actual paper (Cook, 2004) and it turns out the bicycle was symmetrical... and, in fact, entirely virtual.

It's a plot of a computer simulation, rather than records from a real-world physical experiment.

A bicycle is composed of four rigid bodies: the two wheels, the frame, the front fork (the steering column). Each adjacent pair of parts is connected with a joint that allows rotation along a defined axis, and the wheels are connected to the ground by requiring that their lowest point must have zero height and no horizontal motion (no sliding).

So the simulation has a lot of simplifications from reality, and the picture tells us more about the simulation model than it tells us about the real world. It is a pretty picture, though.

Here's the paper reference:

Cook, M. 2004. It takes two neurons to ride a bicycle.

(I couldn't get it from the Cook's Caltech site, but I found a copy elsewhere.)