[AndrewC]

you cannot make angular velocity disappear.

But you just did. "The motion on the one side cancels the motion on the other side."

[Physicist]

But you just did. "The motion on the one side cancels the motion on the other side."

Angular velocity is the angle swept out per unit time, rather than a distance per unit time. It does not cancel, as both "sides" of the object are travelling the the same angular direction, even as they travel in opposite rectilinear directions.