Go Tools->Preferences->Search and then edit the two search engine settings, swapping the content in all of the fields in #1 for those of #2. It's not fun, but Opera doesn't offer the ability to move them up or down in the listings.
I have Opera 9.51. I was wondering if it was possible to rearrange the search engines in the top-right corner. Currently, the oldest search engine is on top, and the newest is on the bottom. Can anyone help.