  Ongoing spectroscopic galaxy surveys like DESI (arXiv:1611.00037,
arXiv:1611.00036) and Euclid (arXiv:2405.13491) will cover unprecedented
volumes with a number of objects large enough to effectively probe clustering
anisotropies through higher-order statistics. In this work, we present an
original and efficient implementation of both a model for the multipole moments
of the anisotropic 3-point correlation function (3PCF) and for their estimator.
To assess the adequacy of our model, we have predicted the anisotropic 3PCF for
a set of dark matter halos at redshift $z = 1$ and compared our prediction with
the 3PCF measurement obtained by applying our estimator to a suite of 298
N-body + 3000 approximate mock halo catalogs at the same redshift. The use of
the anisotropic component of the 3PCF effectively breaks the degeneracy between
the growth rate $f$ and the linear bias $b_1$, allowing us to obtain unbiased
estimates for both parameters from a Euclid-like survey. A joint, full
anisotropic, 2 and 3-point correlation analysis would also allow us to measure
the clustering amplitude $\sigma_8$ along with $f$ and $b_1$ with a precision
of approximately 17%, and to effectively constrain all galaxy biasing
parameters of the model. However, these results largely exploit information
from the isotropic part of the 3PCF signal. The addition of the anisotropic
multipoles tightens parameter estimates by just 5% at most. We argue that this
modest improvement likely reflects the use of a simplified cosmological model
with relatively few free parameters. The 3PCF anisotropic multipoles will prove
useful in reducing the impact of projection effects that affect
high-dimensional cosmological analyses. To support this conjecture, we provide
an example in which an additional source of anisotropy, the Alcock-Paczynski
effect, is added to the system.
