  Gravitational baryogenesis is one of the mechanisms which help us to explore
more about our early universe, especially baryon-anti-baryon asymmetry. As we
know, modified theories of gravity are very successful in describing the
present accelerated scenario of the universe. Therefore, in this letter, we aim
to constrain the generalized torsion-based modified theory of gravity, namely,
$f(T,\mathcal{T})$ gravity with gravitational baryogenesis, where $T$,
$\mathcal{T}$ are the torsion scalar, trace of the energy-momentum tensor,
respectively. For this, we examine how the various Lagrangian forms of
$f(T,\mathcal{T})$ affect the baryogenesis. We also impose the constraints on
the extra degrees of freedom induced by modified theory with the observational
values of the baryon-to-entropy ratio. In addition, we further explore how more
generalized gravitational baryogenesis can attribute in a physically viable and
consistent way to the cosmologies of the modified theory of gravity.
