Let \begin{align} H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}\in M_2(\mathbb C), \quad \mathrm{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix} \in M_4(\mathbb C), \quad U = (H\otimes I_2)\mathrm{CNOT}.\end{align} Then the Bell state measurement is the unitary channel given by $\Phi:\rho \mapsto U\rho U^*$.
The Bell states are \begin{align}\beta_{i,j} = \frac{|0,j\rangle + (-1)^i|1, j\oplus 1\rangle}{\sqrt{2}}, \qquad i,j\in\{0,1\}. \end{align} Let $P_{i,j}$ be the orthogonal projection onto the subspace spanned by $\beta_{i,j}$. Then $\{P_{i,j}:i,j\in\{0,1\}\}$ forms a projective measurement.
I want to understand if the Bell measurement $\Phi$ is the same as the projective measurement $\{P_{i,j}\}$, in the sense whether one can be derived from the other? I thought in this case, the Bell states should be the eigenvectors of $U$, but they are not.
In a more general sense, given a unitary channel, is it possible to construct a projective measurement from it, and vice versa?