  There are at least two ways to encode gravity into geometry: Einstein's
general theory of relativity (GR) for the metric tensor, and teleparallel
gravity, where torsion as opposed to curvature encodes the dynamics of the
gravitational degrees of freedom. The main purpose of the paper is to explore
the relation between loop gravity and teleparallel gravity. We argue that these
two formulations of gravity are related to two different discretizations of the
Einstein--Cartan action, which were studied recently in the literature. The
first discretization leads to the \emph{{loop gravity}} kinematical phase space
where the zero torsion condition is enforced {first} and the other is the
\emph{{dual loop gravity}} kinematical phase space where curvature is imposed
to vanish {first}. Our argument is based on the observation that the GR
first-order Einstein--Cartan action can also be seen as a first-order action
for teleparallel gravity up to a boundary term. The results of our paper
suggest that the \emph{dual loop gravity} framework is a natural discretization
of teleparallel gravity, whereas \emph{loop gravity} is naturally related to
the standard GR metric description.
