When you look at the twisted cord illusion it should seem to you that the lines are bent one way or another, and tend to be bowed near the center of the window. As the stripes reverse their colors, the lines appear to bend in the opposite direction. The lines are straight and parallel.
The twisted cord effect is likely due to orientation-sensitive simple cells in the striate cortex, which interact to combine closely-spaced tilted lines into a single tilted line.
References:
Coren, S. & Girgus, J.S. (1978). Seeing is Deceiving: The Psychology of Visual Illusions. Hillsdale, New Jersey: Lawrence Erlbaum Associates.
Luckiesh, M. (1965). Visual Illusions, Their Causes, Characteristics and Applications. New York: Dover.