= Solution
The specified <time reversal> includes complex conjugating $i\gamma^\mu\partial_\mu$; omitting that operation changes the answer. Put $x_T=(-t,x,y)$, $U_T=\gamma^2$, and $(r_0,r_1,r_2)=(-1,1,1)$. As in <3/c>, the <Dirac adjoint> in the source's transformed bilinear is $\bar\psi'(x)=\bar\psi(x_T)\gamma^2$. The conjugated <Dirac operator> has coefficient matrices
$$
(i\gamma^0)^*=-i\gamma^0,\qquad
(i\gamma^1)^*=+i\gamma^1,\qquad
(i\gamma^2)^*=-i\gamma^2.
$$
Combining these coefficients with the <chain rule> signs and the <Clifford algebra> gives, without summing over $\mu$,
$$
\gamma^2(i\gamma^\mu)^*\gamma^2r_\mu=i\gamma^\mu
\quad(\mu=0,1,2).
$$
For example the time coefficient is $\gamma^2(-i\gamma^0)\gamma^2(-1)=i\gamma^0$, and the second spatial coefficient is $\gamma^2(-i\gamma^2)\gamma^2=i\gamma^2$. Thus the <kinetic term> is invariant. The real <Dirac mass term> again reverses sign since $\bar\psi'\psi'=\bar\psi(\gamma^2)^2\psi=-\bar\psi\psi$. The spacetime integration has unit absolute <Jacobian determinant>, so
$$
\boxed{T:S_m\mapsto S_{-m},\qquad TP:S_m\mapsto S_m.}
$$
Each operation reverses the sign of the mass, while their composition reverses it twice and preserves the <kinetic term>. In an operator formulation the complex conjugation of coefficients expresses the <antiunitary operator> character of <time reversal>; the transformation must not be treated as an ordinary unitary substitution with all $i$'s held fixed. These statements concern the stated free <action>.
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