The purpose of this paper is to introduce the Symmetric Lorentz transformations. These new transformation equations are the foundations of a new theory of relativity called: Symmetric Special Relativity. In this paper several issues are analysed. Firstly, I derive the formula for time dilation. Secondly, I derive the formula for length contraction. Thirdly, I derive a new relativistic velocity composition formula which encompasses part of Einstein's counterpart. Fourthly, I prove that Newton's law of Universal Gravitation is invariant under the new transformation. Fifthly, I prove that the leptobaryonic formula for the fine-structure constant is invariant under the transformation. Lastly, I mention that the de Broglie formula is not invariant under the transformation (the proof is not included in this article). It seems both the Lorentz transformation and the Symmetric Lorentz transformation can indicate whether a given classical mathematical description of nature is relatively accurate in its respective domain (e.g. Maxwell's equations are invariant under a Lorentz transformation while Newton's Gravity Law is invariant under a Symmetric Lorentz transformation). Therefore, the two transformations complement each other. One marvellous advantage of having “complementary” transformations is that we can take advantage of them.