If you stuff a Mask inside a bag and get back to town,the Mask will remain in the bag instead of showing up on the post exploration screen and Training ground.
I noticed the mask thing and it also applies to gems, blacksmith patterns, and anything else which is otherwise automatically removed upon going back to town. I also had a key from the fourth area disappear on me when I had it in my inventory upon going back to town, but I was able to keep it a second time around by putting it in a bag. I have no idea if it's meant to disappear or not.