  We investigate in detail the implications of the constant-roll condition on
the inflationary era of a scalar field coupled to a teleparallel $f(T)$
gravity. The resulting cosmological equations constitute a reconstruction
technique which enables us to find either the $f(T)$ gravity which corresponds
to a given cosmological evolution, or the Hubble rate of the cosmological
evolution generated by a fixed $f(T)$ gravity. We also analyze in some detail
the phase space of the constant-roll teleparallel gravity and we discuss the
physical significance of the resulting fixed points and trajectories. Also we
calculate the observational indices of a theory with given $f(T)$ gravity, and
we discuss all the implications of the constant-roll condition on these. As we
demonstrate, the resulting theory can be compatible with the current
observational data, for a wide range of values of the free parameters of the
theory.
