I think the light ray travelling directly with/against the flow of the aether should be the expected ray to be delayed a bit, causing the change in interference patterns.
Reason why is that it has to travel at a slower rate in theory if and only if the aether existed, when you look at the derivation.
However, the main focus here is the interference pattern produced. When rotated, if the interference patterns are different in comparison, then one light is arriving at a later time than the other. However if they are the same patterns both times exactly (after and before rotating 90 degrees), then what we can deduce is that both beams are travelling the same distance relative to before and after (NOTE: NOT relative to each other as beams, but relative to the cases before and after) regardless of the existence of an aether. Hence, this explains why light does not require a medium to propagate within. This deems the existence of the aether void and of no point now.