Full symmetry

A polyomino has all symmetries of the square or full symmetry if it is equal to its reflection over any axis or diagonal line, or any 90˚ rotation. This symmetry is represented by the dihedral group $D_4$, the symmetry group of the square.

The smallest polyomino that has this symmetry is the monomino (a single square). This type of symmetry is only possible if the polyomino has size $4k$ or $4k+1$ for some nonnegative $k$.

Polyomino list