The mesh on the left is before morphing; the mesh on the right is after morphing.  The icon for 1-plane symmetry is a rectangle perpendicular to the symmetry system's selected axis.  You can think of this rectangle as a mirror. The highlighted handle is moved.  Note how only the handle at the lower left has been selected and how the handle on the upper left is automatically moved symmetrically.  This type of symmetry is very useful for a wide variety of symmetric models.