= Solution
Use the normalized <Fricke involution> $W_N=f\mapsto f|_k\left(\begin{smallmatrix}0&-1\\N&0\end{smallmatrix}\right)$, and let $D_\ell=\left(\begin{smallmatrix}\ell&0\\0&1\end{smallmatrix}\right)$. Choose integers $m,n$ with $m\ell-Nn=1$. The matrix $\gamma=\left(\begin{smallmatrix}1&n\\N&m\ell\end{smallmatrix}\right)$ lies in $\Gamma_1(N)$, and $\gamma'=\left(\begin{smallmatrix}\ell&n\\N&m\end{smallmatrix}\right)$ has determinant one and lies in $\Gamma_0(N)$, with $m\equiv\ell^{-1}\pmod N$. The displayed identity in the PDF reads $D_\ell\gamma^{-1}=\operatorname{diag}(1,\ell)\gamma'$. Taking double cosets therefore identifies the reversed correspondence with the <diamond operator> $\langle\ell^{-1}\rangle$ times the original one. Conjugation by $W_N$ reverses the two diagonal entries, so
$$
\boxed{W_NT_\ell W_N^{-1}=\langle\ell^{-1}\rangle T_\ell.}
$$
The same finite-correspondence change of variables in the <Petersson inner product> transfers a matrix to its inverse and yields $T_\ell^*=\langle\ell^{-1}\rangle T_\ell$. The <diamond operators> commute with $T_\ell$, so $T_\ell T_\ell^*=T_\ell^*T_\ell$. Thus $T_\ell$ is a <normal operator>. Multiplying $W_N$ by a nonzero scalar does not change the conjugation formula.
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