Let $w$ be a primitive $2m$th root of unity. Then the sequence generated by $$x_{n+2}=\frac{w^4x_n-(w^3+w^2)x_{n+1}-wx_nx_{n+1}}{-w-(w^3+1)x_n+w^2x_{n+1}}$$ appears to have period $2m$ for almost all initial terms.
I can prove this for (very) small $m$ and check it numerically for other values of $m$. I should be grateful for any ideas regarding a general proof.
The equation can be written as an equality of a homogeneous quadratic and a linear polynomial: $$wx_nx_{n+1}-(w^3+1)x_nx_{n+2}+w^2x_{n+1}x_{n+2}=w^4x_n-(w^3+w^2)x_{n+1}+wx_{n+2}.$$ Sequences with $x_{n+1}-wx_n=0$ send both the quadratic part and linear part to zero. This suggests writing the equation in the form $$w^2(x_{n+2}+w)(x_{n+1}-wx_n)=(x_n+w)(x_{n+2}-wx_{n+1}).$$ Restricting to sequences not of the form $x_i=Aw^i,$ this is equivalent to $$w^2\frac{(x_{n+2}+w)(x_{n+1}+w)}{x_{n+2}-wx_{n+1}}=\frac{(x_n+w)(x_{n+1}+w)}{x_{n+1}-wx_n}.$$
This shows that the numbers $$y_n=\frac{(x_n+w)(x_{n+1}+w)}{x_{n+1}-wx_n}$$ satisfy $y_{n+1}=w^{-2}y_n,$ and are therefore $2m$-periodic.
If the $y_n$ are fixed, the map $x_n\mapsto x_{n+1}$ can be considered as a Möbius transformation: $$x_{n+1}=\frac{(wy_n+w)x_n+w^2}{-x_n+(y_n-w)}.$$
and the problem is reduced to showing that the composition $x_0\mapsto x_1\mapsto \cdots\mapsto x_{m}$ is an involution - doing it twice ends up at $x_0.$ This map is represented by the matrix product $A({y_{n-1}})\cdots A(y_0)$ where $$A(y)=y\begin{pmatrix}w&0\\0&1\end{pmatrix}+\begin{pmatrix}w&w^2\\-1&-w\end{pmatrix}.$$
Let $p(y)=\operatorname{tr}(A({yw^{-2(m-1)}})\cdots A(yw^{-2})A(y)).$ Then $p$ is a degree-$m$ polynomial in $y.$ By conjugation invariance of trace we have $p(y)=p(yw^2),$ which means $p$ is of the form $Ay^m+B,$ where $A=w^{-2(m-1)+\dots-2-0}\operatorname{tr}\begin{pmatrix}w^m&0\\0&1\end{pmatrix}=0$ and $B=\operatorname{tr}\begin{pmatrix}w&w^2\\-1&-w\end{pmatrix}^m=0.$ So $p=0,$ which means $x_0\mapsto x_m$ is an involution as required.
This argument shows that most choices of $x_n,x_{n+1}$ give period $2m.$ It is also possible for a sequence to have period $m.$ There will generally be two such sequences for each $y_0,$ corresponding to the fixed points of the Möbius transformation $x_0\mapsto x_m.$