Why do linear phase filters have symmetric impulse responses?
Actually, I think I see why.

$$X(j\Omega) = |X(j\Omega)|e^{-j\theta(\Omega)}$$

$|X(j\Omega)|$ is purely real, and therefore if we take the IFT it is even and symmetric.

$\theta(\Omega)= a\Omega$ since the phase is linear, so $e^{-ja\Omega}$ merely shifts the corresponding even and symmetric magnitude in the time domain, so the resulting impulse response is symmetric about $a$.