Can I keep the Rueful Axe and get Clavicus Vile's Mask?
In this particular quest, you can not get both items as reward.
You either kill Barbas (poor mutt) and keep the Rueful Axe.
Hand the axe to Clavicus Vile and receive his true daedric artifact, viz. the Masque of Clavicus Vile.
Important thing to note here is that only the Masque of Clavicus Vile counts towards the Oblivion Walker achievement.
I did try to be clever and decided not to kill Barbas before he turned to stone. This glitched my game worse than ever. Clavicus Vile's statue disappeared.
Please see this page on UESP for further information.
drat. I could have sworn I read somewhere it was possible. oh well, guess that means I don't have to reload a past save to redo the quest at least.
Its no glitch, its just some kind of odd reaction because most player don't try that. You cannot get both items.