When is a system's frequency response symmetric?
This property applies to any system whose impulse response is real-valued (as opposed to complex-valued).

For a real-valued function of time \$f(x)\$, the Fourier transform \$F(\omega)\$ has the property

$$F(\omega)=F^\star{}(-\omega),$$

where \${}^\star\$ indicates the complex conjugate. Since \$\|z^\star\|=\|z\|\$, the magnitude spectrum of a real-valued function is symmetric.