  In this paper we perform the reconstruction scheme of the gravitational
action within $f(T,\mathcal{T})$ gravity, where $T$ and $\mathcal{T}$ denote
the torsion scalar and the trace of the energy momentum tensor, respectively.
We particularly focus our attention on the case where the algebraic function
$f(T,\mathcal{T})$ is decomposed as a sum of two functions $f_1(T)$ and
$f_2(\mathcal{T})$, i.e, $f(T,\mathcal{T})=f_{1}(T)+f_{2}(\mathcal{T})$. The
description is essentially based on the scale factor and then, we consider two
interesting and realistic expressions of this parameter and reconstruct the
action corresponding to each phase of the universe. Our results show that some
$f(T,\mathcal{T})$ models are able to describe the evolution of the universe
from the inflation phase to the late time dark energy dominated phase.
