I had same problem post upgrade to iOS 6.1.3
I went to
[Setting] [General] and set the
[Use_Side_Switch_To] option from [Rotation ...] to [Mute]
Checked switch by manually flipping the button on iPad cover (next to volume) back and forth and seeing images of mute actions appear sceen centre
But on checking further not all apps rotated automatically ... so ...
Manually closed all open apps by:
Screen button x2 in quick succession bottom bar appears with all active apps
Touch and hold on of these apps in this bar untill the x appears on them and the wobble
Close each one down by touching x
Then
Rebooted ( ... just like a Microsoft machine :-)
Now all working working ... boy you really miss that feature when its not functioning ...
Dont know if all steps relevant but this is what worked for me ...
... best of luck
Message was edited by: citizenpips