Why are Fresnel coefficients not symmetric?

Question

When talking about reflection, we have the following coefficients for the electric field: $$r_{\perp}=\frac{n_1\cos(i)-n_2\cos(t)}{n_1\cos(i)+n_2\cos(t)} \\ r_{\parallel}=\frac{n_2\cos(i)-n_1\cos(t)}{n_2\cos(i)+n_1\cos(t)}$$ But when the incident angle becomes zero, the transmission angle will also become zero ($i=0\ \&\ t=0$) and then we have: $$r_{\perp}=\frac{n_1-n_2}{n_1+n_2}\\ r_{\parallel}=\frac{n_2-n_1}{n_2+n_1}$$ Why are those not symmetric? When the incident angle is zero, there should be no distinction between perpendicular and parallel to the incident plane because we can rotate that plane and it would still be incident plane. Why do the Fresnel equations say otherwise?

Answer 1

In the original derivation of these formulae, we assume that the reflected parallel wave is in antiphase which means that when using the coefficient $r_{\parallel}$, we should get the negative component.