What is this icon on my iPad Pro - blue with syncing arrows around a gear wheel?

You're very welcome.

In summary, the icon is benign - and is nothing for which you should have any concern. Hopefully Apple will eventually get around to documenting this and other unlisted status icons.

The icon indicates that your iPad is synchronising with "something".

Yours is one of many reports seen in recent weeks. At present, pending confirmation by Apple, this appearance of the icon is believed to have some relationship to post-update synchronisation tasks performed by your iPad after updating to iPadOS 18.x - or the icon appears while a download is being performed as a background task.

Apple have not yet provided a definitive explanation.

Apple's documentation is often slow to catch-up with new features. Here is another status icon - that appears briefly when connecting some USB-C hubs - that Apple have yet to add to their documentation: