What is importance of the conjugate reciprocal and none reciprocal zeros in FIR Type 2 Filters?
I would like to provide a short answer.

Consider

$$ h[n] = \sum_{k=0}^{M} b_k \delta[n-k] ~~~~\longleftrightarrow ~~~~H(z) =  \sum_{k=0}^{M} b_k z^{-k} $$

Left: LTI FIR filter impulse response coefficients $b_k$, and right: its Z-Transform.

If the FIR filter is real; i.e., coefficients $b_k$ are real, then the Z-transform is a polynomial of real coefficients, and the roots of such polynomials are either real complex-conjugate pairs. This explains why a real FIR filter's Z-domain poles &amp; zeros are in conjugate pairs when they are complex.

To explain why, also, they are in reciprocals, we need to consider linear phase property (without losing generality) : $h[n] = h[-n]$; then since the Z-transform of $h[-n]$ is $H(1/z)$; we have the following :

$$ h[n] = h[-n]  ~~~\implies ~~~ H(z) = H(1/z) $$

And this means that if $z_0$ is a zero (or pole) of $H(z)$; i.e., $H(z_0) = 0$ then its reciprocal $1/z_0$ is also a zero of $H(z)$.

To sum up; for a real and linear phase FIR filter, if $z_0$ is a zero of $H(z)$ then $z_0^*$, $1/z_0$ and $1/z_0^*$ are also zeros; hence zeros and poles are in pairs of four when they are complex.