  We present an algorithm enabling computation of the anisotropic
redshift-space galaxy 3-point correlation function (3PCF) scaling as $N^2$,
with $N$ the number of galaxies. Our previous work showed how to compute the
isotropic 3PCF with this scaling by expanding the radially-binned density field
around each galaxy in the survey into spherical harmonics and combining these
coefficients to form multipole moments. The $N^2$ scaling occurred because this
approach never explicitly required the relative angle between a galaxy pair
about the primary galaxy. Here we generalize this work, demonstrating that in
the presence of azimuthally-symmetric anisotropy produced by redshift-space
distortions (RSD) the 3PCF can be described by two triangle side lengths, two
independent total angular momenta, and a spin. This basis for the anisotropic
3PCF allows its computation with negligible additional work over the isotropic
3PCF. We also present the covariance matrix of the anisotropic 3PCF measured in
this basis. Our algorithm tracks the full 5-D redshift-space 3PCF, uses an
accurate line of sight to 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 RSD in current and upcoming large-scale redshift surveys.
