We present an algorithm enabling computation of the anisotropicredshift-space galaxy three-point correlation function (3PCF) scaling asN2, with N the number of galaxies. Our previous work showedhow to compute the isotropic 3PCF with this scaling by expanding theradially binned density field around each galaxy in the survey intospherical harmonics and combining these coefficients to form multipolemoments. The N2 scaling occurred because this approach neverexplicitly required the relative angle between a galaxy pair about theprimary galaxy. Here, we generalize this work, demonstrating that in thepresence of azimuthally symmetric anisotropy produced by redshift-spacedistortions (RSD), the 3PCF can be described by two triangle sidelengths, two independent total angular momenta, and a spin. This basisfor the anisotropic 3PCF allows its computation with negligibleadditional work over the isotropic 3PCF. We also present the covariancematrix of the anisotropic 3PCF measured in this basis. Our algorithmtracks the full 5D redshift-space 3PCF, uses an accurate line of sightto each triplet, is exact in angle, and easily handles edge correction.It will enable use of the anisotropic large-scale 3PCF as a probe of RSDin current and upcoming large-scale redshift surveys.