A unified symmetrical theory of DNA sequences has been established based on the basic symmetry of the DNA bases. It is shown that the symmetry of DNA sequences is inherently related to that of a cube and its inscribed regular tetrahedron. A DNA group is defined as a particular alternating group of order 4, in which the permuted objects are four bases. The symmetry of DNA sequences is described by the DNA group which is isomorphic to the tetrahedral group. The matrix representation for the DNA group has been obtained, and used to establish the relationships between the transforms of bases and the rotations of the tetrahedron. It is found that any DNA sequence can be uniquely described by three independent distributions, i.e., the distributions of the bases of purine/pyrimidine, of amino group/keto group and of strong/weak hydrogen bonds along the sequence. The three distributions are invariant in some sense under the transforms of the DNA group, indicating that the three distributions are inherent for the sequence. The mathematical format of the theory lays a foundation for further development. The applications of the theory to analyse some DNA sequences are presented.