When you look into a concave mirror you're looking at an image of yourself. The position of the image is described in this article on the mirror equation.
When you are farther away from the mirror than the focal point a real image is formed between the mirror and it's focal point. This image is inverted. When you get closer to the mirror than the focal point a virtual image is formed behind the mirror and this image is not inverted. That's why the image flips as you get closer. You'll also find the real image is smaller than the object while the virtual image is bigger than the object.
A quick Google found this article that shows the ray diagrams for real and virtual images.
Beginners to optics find the idea of virtual images confusing. A real image can be seen on a screen i.e. if you put a piece of paper in the position of the image you'll see the image on the paper. With a virtual image the light rays never come to a focus so there is no place you can put a piece of paper to see the image. However your eye contains a lens and can bring the diverging light rays to a focus on your retina. Hence your eye can see a virtual image even though it couldn't be projected onto the sheet of paper.