The modified SWAP exchanges adjacent Majorana pairs, with its scalar phase retained

For every natural number $n$, every family $c_0,\ldots,c_{2n-1}$ of Hermitian matrices on $n$-bit strings satisfying $c_rc_s+c_sc_r=2\delta_{rs}I$, and every injection $w:\{0,1\}\hookrightarrow\{0,\ldots,n-1\}$ with $w(1)=w(0)+1$, three assertions hold. Write $c_{a,b}=c_{2w(a)+b}$ and $W_c=\exp\!\left(-\frac{\pi}{4}[-c_{0,0}c_{1,1}+c_{0,1}c_{1,0}+c_{0,0}c_{0,1}+c_{1,0}c_{1,1}]\right)$. First, $W_c^\dagger c_{a,b}W_c=c_{1-a,b}$ for every $a\in\{0,1\}$ and Boolean $b$. Second, for the particular family $j_{2k}=Z^{\otimes k}\otimes X\otimes I^{\otimes(n-k-1)}$ and $j_{2k+1}=Z^{\otimes k}\otimes Y\otimes I^{\otimes(n-k-1)}$, one has $W_j=-iE_w(K)$, where $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$, $Z=\operatorname{diag}(1,-1)$, and $K$ is the two-bit matrix acting as $Z$ on the even-parity sector indexed by the first bit and as $X$ on the odd-parity sector, with zero entries between the sectors. Here $E_w(K)_{x,y}=K_{x\circ w,y\circ w}$ when $x,y$ agree outside the image of $w$, and is zero otherwise. Third, this same $K$ equals controlled-$Z$ multiplied by the swap matrix, after identifying a two-bit string with its ordered pair of bits; explicitly, $K_{x,y}=(-1)^{y_0y_1}$ if $(x_0,x_1)=(y_1,y_0)$, and $K_{x,y}=0$ otherwise. The wire injection excludes $n<2$.