The analysis of the impact of spin-orbit coupling (SOC) on the Kondo state has generated considerable controversy, mainly regarding the dependence of the Kondo temperatureon SOC strength. Here, we study the one-dimensional (1D) single impurity Anderson model (SIAM) subjected to Rashba () and Dresselhaus () SOC. It is shown that, due to time-reversal symmetry, the hybridization function between impurity and quantum wire is diagonal and spin independent (as it is the case for the zero-SOC SIAM), thus the finite-SOC SIAM has a Kondo ground state similar to that for the zero-SOC SIAM. This similarity allows the use of the Haldane expression for, with parameters renormalized by SOC, which are calculated through a physically motivated change of basis. Analytic results for the parameters of the SOC-renormalized Haldane expression are obtained, facilitating the analysis of the SOC effect over. It is found that SOC acting in the quantum wire exponentially decreaseswhile SOC at the impurity exponentially increases it. These analytical results are fully supported by calculations using the numerical renormalization group (NRG), applied to the wide-band regime, and the projector operator approach, applied to the infinite-regime. Literature results, using quantum Monte Carlo, for a system with Fermi energy near the bottom of the band, are qualitatively reproduced, using NRG. In addition, it is shown that the 1D SOC SIAM for arbitraryanddisplays a persistent spin helix SU(2) symmetry similar to the one for a 2D Fermi sea with the restriction=.