Degeneracy is a salient feature of genetic codes, because there are more codons than amino acids. The conventional table for genetic codes suffers from an inability of illustrating a symmetrical nature among genetic base codes. In fact, because the conventional wisdom avoids the question, there is little agreement as to whether the symmetrical nature actually even exists. A better understanding of symmetry and an appreciation for its essential role in the genetic code formation can improve our understanding of nature coding processes. Thus, it is worth formulating a new integrated symmetrical table for genetic codes, which is presented in this paper. It could be very useful to understand the Nobel laureate Crick wobble hypothesis: how one transfer ribonucleic acid can recognize two or more synonymous codons, which is an unsolved fundamental question in biological science.