Hi there Julie,
It sounds like you have turned off rotation lock but your screen still will not rotate. If I understand correctly you have backed up the iPad to iTunes and updated it but the issue persists. This can be either a software or a hardware issue, and these steps will help us rule out software. Start by quitting all the running apps on the iPad:
Force an app to close in iOS
https://support.apple.com/en-us/HT201330
1. Press the Home button two times quickly. You'll see small previews of your recently used apps.
2. Swipe left to find the app you want to close.
3. Swipe up on the app's preview to close it.
When you have done that force restart the device with the Home and Sleep/Wake buttons the way this article describes, and test the issue again:
iOS: Turning off and on (restarting) and resetting
http://support.apple.com/en-us/HT201559
If the issue persists, backup your device to iTunes, then restore it as new and verify that it works.
How to erase your iOS device and then set it up as a new device or restore it from backups
https://support.apple.com/en-us/HT204686
If it does, then restore your backup to either verify it still works and the software just needed reinstalled, or isolate the issue to the backup file itself.
Let us know how that goes! Thank you for using Apple Support Communities.