I bought the most adorable pink/white/& grey reversible plaid jacket. Any suggestions how I can go about embroidering on it?

About the only thing I could think of, if you didn't want to seperate the two layers, would be to choose something like an FSL design and use the same color bobbin thread as you do, top thread. Then you would have an identical design, but it would be on the opposite side when you reverse the jacket.

Is it 2 layers? Can you take it apart at a side seam, embroider it, and sew it back up?
If it's a reversible fabric that doesn't have 2 layers you could embroider on something else (like organza), cut it out, and hand stitch it on to one side of the jacket.
Maybe some other Cuties will have more ideas for you.