Yes, doing something that does NOT speak to me because it's important to the other person can be tremendously meaningful to me. I think I'm seeing obedience a little bit differently, though. To me, the very act of obeying can be very meaningful. If I only participate insofar as an action resonates with me, then the interaction is less about submission than about mutual gratification. Participating despite the fact that something DOESN'T resonate with me invests the action with a certain kind of surrender.
But, not to get too fine-grained here, then it's not the act itself that has the significance - the act is meaningless. It's the obedience, the fact of wrestling with and twisting and reshaping the content of my submissive intuitions, that has the significance. So there's still a very real and meaningful event occurring.
Obviously it just comes down to the motivation. I can do something because it's clearly a submissive act that resonates with me, or I can behave out of trust or obedience or respect, but I can also perform the same actions just for fun, or as an indulgence to the other person. It's the last category that makes me feel, well, ironic, or inauthentic, or detached, but it certainly also makes me feel much more in control.