I have a morph creature in play that was turned face down by Ixidron's ETB trigger, meaning that my opponent knows what that creature is. If I then play another morph creature face down, am I allowed to (for example) shuffle the two creatures and put them back on the table, so that my opponent does not know which one is the new one?
No.
All other players need to know the relative order in which your face down creatures entered the battlefield. If one or more of them were face up and turned face down by an effect like Ixidron, they have the right to know which face down creature was which face up creature.












