One possible way to linearize such a constraint would be by dividing this expression into four parts and then linearizing each of which separately.
$$ Iff \quad (w=1) \rightarrow (x \rightarrow y) \quad (\text{Part-1})$$
$$ Iff \quad (w=1) \rightarrow (y \rightarrow x) \quad (\text{Part-2})$$
$$ Iff \quad (x \rightarrow y) \rightarrow (w=1) \quad (\text{Part-3})$$
$$ Iff \quad (y \rightarrow x) \rightarrow (w=1) \quad (\text{Part-4})$$
Then, the first one can be translated as:
$$ (w) \rightarrow (\lnot x \lor y)$$
$$ \lnot(w) \lor (\lnot x \lor y)$$
$$ \lnot w \lor \lnot x \lor y$$
$$ (1-w) + (1-x) + y \geq 1 \quad (\text{Part-1})$$
The second part would be the same as the one and the result is:
$$ (1-w) + (1-y) + x \geq 1 \quad (\text{Part-2})$$
The third part can be linearized as:
$$ (\lnot x \lor y) \rightarrow (w)$$
$$ \lnot (\lnot x \lor y) \lor(w)$$
$$ (x \land \lnot y) \lor (w)$$
$$ (x \lor w) \bigwedge (\lnot y \lor w)$$
$$ (x + w \geq 1) \bigwedge ((1-y) + w \geq 1) \quad (\text{Part-3})$$
$$ (y + w \geq 1) \bigwedge ((1-x) + w \geq 1) \quad (\text{Part-4})$$
Also, the third and fourth parts can already be written as:
$$ \lnot ((\lnot x \lor y) \land (\lnot y \lor x)) \lor w$$
$$ (x + y + w \geq 1) \bigwedge (w \geq x + y - 1) \quad (\text{Part-(3,4)})$$