Time-reversal reciprocity of the scattering matrix ($⟨f|S|i⟩=⟨\hat{T}i|S|\hat{T}f⟩$)

If a Hermitian interaction-picture Hamiltonian satisfies $\hat{T}V(t){\hat{T}}^{-1}=V(-t)$ for an antiunitary operator, its Dyson series conjugates $(-i{)}^n$ to $i^n$. The substitution $t_j\mapsto -t_j$ converts ordered integration to anti-ordered integration, giving $\hat{T}S{\hat{T}}^{-1}=S^{†}$. Antiunitarity then gives the displayed reciprocity: the time-reversed process has the reversed outgoing state as its incoming state and vice versa. It does not merely reverse the momenta while leaving their incoming/outgoing roles fixed.