Let's say I have a square, rectangle, or circle. By default, the center point of will be positioned exactly in the middle. It is so because the binding box is same as the shape. It can be viewed by the Attributes Panel and clicking "Show Center".
I also have some shapes which are irregular. If I look at the center point, it is positioned in the middle but the middle is relative to the bounding box of that shape. How would I go about selecting and editing the center point in such a way that it sits exactly in the center, horizontal and vertical relative to the shape and not the bounding box.
In the below image, on the top there is a square, rectangle, and circle. The center point is exactly in the center.
Below that there is a chicken, cow, and fish. The center point are not in the middle.
The center point is always in the center of the object. It is not associated with the bounding box. You can turn off the Bounding box and retain the center point.
The object's (shape's) outer width and outer height are used to draw the center point as well as its bounding box. What you see is the exact center of your shapes. Moving it would place it off-center.
With symmetrical shapes, such as the rectangles and circle, it appears to be centered more. However it's just as centered in the irregular shapes of the chicken, cow, and fish. The difference is that every associated path in the irregular shapes is not an equal distance from the center of the shape as it is in the symmetrical shapes.
To illustrate more... see the attached image. It clearly shows the center point is in equidistant from either side. Green rectangle is merely a copy of the red rectangle.
However, all of this is irrelevant since you can not edit or alter the center point beyond showing and hiding it.
Unlike Flash, Adobe Illustrator doesn't save anchor point data with a drawing object. The only exception is with symbols. A workaround that I use is to draw a circle around the object and making it transparent. Basically, the center of the object will not change, but it will achieve the desired effect. The example attached shows that: if you try to rotate the green star, it will not rotated using the visual center of the object. But if you try to rotate the yellow star, which is grouped with the circle, it will.
Hope it helps!
Now to create the symbol: