How much is phase delay of ring resonator fed by two orthogonal feed lines? 180 deg?

[Terminator3]

I read papers on ring resonators and filters. There is something keep me dizzy:
if i take simple ring resonator, perimeter = wavelength, and phase delay would be wavelength/2, so 180 deg (i hope it is right?) As feed lines are at left and right sides of the ring, wave will propagate 180 deg on both sides (assume coupling gaps are small). Nulls at 90.

And here is the part i can't get: if ring fed by two orthogonal feed lines (for example, left and bottom), then there would be 2 paths: 90deg and 270deg. From few papers i found out that S21 for n*F=1*F would be very low, only n=2*F (wavelength/2) is usable.
So for frequency F (some wavelength Lambda) we can use ring resonator of double perimeter 2*Lambda (F/2).

Summary:
feed signal: Lambda/2
ring parameter: Labda
orthogonal feed lines gives 90 deg at Lambda, but signal is Lambda/2, so phase shift would be 180.
but what happens with the upper part? 270*2=540 deg; 540-360=180 deg. Again 180?

Does it means, that ring resonator designed for F with orthogonal feed can be used at n*F frequencies, where n=2,4,6,...?

Thanks