Bloch electrons in a magnetic field: Why does chaos send electrons the hard way?

We find that a 2D periodic potential, with different modulation amplitudes in the x and y directions, and a perpendicular magnetic field may lead to a transition to electron transport along the direction of stronger modulation and to localization in the direction of weaker modulation. In the experimentally accessible regime we relate this new quantum transport phenomenon to avoided band crossings due to classical chaos.