An important, relatively frequent special form of magic squares are *symmetrical* squares, where all pairs of cells diametrically equidistant from the center have the same sum n^{2} + 1. These number pairs are said to be *complementary*.

46 | 19 | 41 | 14 | 29 | 2 | 24 |

17 | 39 | 12 | 34 | 7 | 22 | 44 |

37 | 10 | 32 | 5 | 27 | 49 | 15 |

8 | 30 | 3 | 25 | 47 | 20 | 42 |

35 | 1 | 23 | 45 | 18 | 40 | 13 |

6 | 28 | 43 | 16 | 38 | 11 | 33 |

26 | 48 | 21 | 36 | 9 | 31 | 4 |

Symmetrical magic squares have been intensively investigated. Some important results are given here:

- The only magic square of the third order is symmetrical.
- The center cell of an odd symmetrical square is always equal to the middle number of the series.
- There are 48 symmetrical squares of fourth order.
- The minimum order for a magic square that is both symmetrical and pandiagonal, is n=5.
- There is
*no*symmetrical square of single-even order. - Each symmetrical magic square is also semi-pandiagonal. But the converse is not true: not every semi-pandiagonal square is also symmetrical.