How is going around the circle once in each direction homotopic to a point? Two paths are homotopic if one can be continuously deformed to the other, right? So I've been told that the fundamental group of the circle is isomorphic to the integers, since you can't deform e.g., the path that goes around once clockwise to that which goes around twice clockwise, since the disc gets in the way, so it's generated by the the clockwise and counterclockwise paths. But that seems to imply that if you go around once clockwise and once counterclockwise, you get a path that can be continuously deformed to the base point? How?

As @Ben says, the key is that the middle point, where you switch from finishing the first loop to starting the second loop, is able to move. Here's some bad pictures to feed your intuition a bit.

If you wrap a rubber band once around a post and let go, it will stay around the post:

![enter image description here](

On the other hand, if you wrap it once, then wrap it the other way, you get this (and when you let go, the band will snap back and have nothing to do with the post):

![enter image description here](