Using symmetry in your composition helps to convey a sense of balance, stability and, in some cases, boredom. Symmetry is when the two halves of a frame match each other almost exactly. This is different from creating a sense of balance. In a balanced shot, the props or characters on one side of the frame might balance out those on the other side but they may not necessarily be symmetrical.
Towards the beginning of Equilibrium (2002), director Kurt Wimmer establishes a society that is devoid of emotion. Throughout this sequence, he frequently uses symmetrical shots to achieve this, including symmetrical shots of men and women in colourless grey clothing sitting in pews, establishing shots of the city that are perfectly symmetrical and balanced. This repeated use of symmetry is a crucial part of establishing this emotionless society for the audience.
Similarly, in the opening of Garden State (2002), Zach Braff uses centre framing and symmetry to convey to the audience that his central character’s life is completely dull and without meaning. The character’s introduction features a series of three perfectly symmetrical and centred overshots of Andrew Largeman (Zach Braff) laying in bed as he listens to a message informing him that his mother has died.
Need to convey balance, stability or perhaps a little boredom? Look for opportunities to use symmetry in your composition.