A real image is seen when light rays from an object are intercepted directly by the eye; the light must reach the eye along a straight-line path without being reflected. This is not the case with a flat mirror.
An image in a flat mirror is a "virtual" image" because the light rays are reflected. The virtual image is formed at the "apparent" position from which light reaches the eye, which is inside the mirror at the same distance the object is to the mirror.
A mirror image is actually reversed front-to-back, not left-to-right. This is because the virtual image we see appears to originate inside the mirror, but the "virtual object" is facing away from the viewer, not toward the viewer.