A sprite's costume center, also called the rotation center, is considered the center of the sprite for technical purposes. Each costume has its own center. By default, it is in the mathematical center of the costume.
When changing direction, the center of rotation is the costume center. If the costume center is not in the center of the costume, this can give the illusion of rotating around an arbitrary point.
In the Paint Editor, there is an option to change the costume center in the upper right hand corner. Clicking the cross-hair image will reveal crosshairs on the screen that follow the cursor, allowing the user to designate a new center.