= Solution
Take the half-turn about the $z$-axis, represented by the <rotation matrix>
$$
\boxed{M=\begin{pmatrix}-1&0&0\\0&-1&0\\0&0&1\end{pmatrix}.}
$$
It is an <orthogonal matrix> with <determinant> $1$, and it fixes the $z$-axis while reversing both perpendicular coordinates. Direct substitution gives $Ma=b$, $Mb=a$, $Mc=d$ and $Md=c$. A <linear map> that permutes the four vertices also preserves their <convex hull>, which is the solid <tetrahedron>. Hence this <rotation matrix> leaves $T$ invariant. Its axis joins the midpoints of the two opposite edges $ab$ and $cd$.
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