  Time-reversal-odd ($\mathcal{T}$-odd) nonlinear current response has been
theoretically proposed and experimentally confirmed recently. However, the role
of disorder scattering in the response, especially whether it contributes to
the $\sigma_{xx}$-independent term, has not been clarified. In this work, we
derive a general scaling law for this effect, which accounts for multiple
scattering sources. We show that the nonlinear conductivity is generally a
quartic function in $\sigma_{xx}$. Besides intrinsic contribution, extrinsic
contributions from scattering also enter the zeroth order term, and their
values can be comparable to or even larger than the intrinsic one. Terms beyond
zeroth order are all extrinsic. Cubic and quartic terms must involve skew
scattering and they signal competition between at least two scattering sources.
The behavior of zeroth order extrinsic terms is explicitly demonstrated in a
Dirac model. Our finding reveals the significant role of disorder scattering in
$\mathcal{T}$-odd nonlinear transport, and establishes a foundation for
analyzing experimental result.
