While the rule of thirds keeps you from placing subjects into the center of the frame, sometimes it's just what you might need.

Symmetry

Using symmetry is very straight forward in photography. If you find symmetry in what you are shooting it's often best to emphasize it by placing it in the center.

Composing our image that way creates balance.

A symmetry line splits our subject into two exact halves. If those halves are identical you can call that object a symmetric object.

Symmetry is an often over-looked tool. It allows us to create an harmonious image with aesthetically pleasing balance and proportions. It's not by chance most classical buildings use symmetry in their facades. From ancient times until today, symmetry remains a key element in design. From buildings to cars, phones to microchips.

Perhaps our fondness to symmetry is explained by ourselves: the human body.

While photographing it's the angle we shoot (how we place our subjects in the frame) at that determines how much we show the symmetry of our subjects.
Lines of symmetry can be found anywhere and in various form, but the most prevalent are:

Vertical Line of Symmetry : If an object’s line of symmetry is perpendicular to the horizon line, it has a vertical line of symmetry.
Horizontal Line of Symmetry : If an object’s line of symmetry is parallel to the horizon line, it has a horizontal line of symmetry.