Poles of the transfer function in Z Transform
The transfer function $H(z)$ is the $\mathcal{Z}$-transform of the impulse response $h[n]$. If $h[n]$ is real-valued, i.e., if $h[n]=h^*[n]$ we have

$$H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n}=\sum_{n=-\infty}^{\infty}h^*[n]z^{-n}=\left[H(z^*)\right]^*\tag{1}$$

So if $z_1$ is a pole of $H(z)$, then $z_1^*$ must also be a pole.

If furthermore $h[n]$ is symmetric, i.e., if $h[n]=h[-n]$ we have

$$H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n}=\sum_{n=-\infty}^{\infty}h[-n]z^{-n}=\sum_{n=-\infty}^{\infty}h[n]z^{n}=H\left(\frac{1}{z}\right)\tag{2}$$

which means that if $z_1$ is a pole, then $1/z_1$ must also be a pole.

In sum, if $h[n]$ is real-valued and symmetric you get for each complex-valued pole $z_1$ three additional poles at $z_1^*$, $1/z_1$, and $1/z_1^*$