The word INORY it self is a simple anagram on IRONY.
Inory is the complete opposite of being sarcastic and/or ironic. This word may come in handy for persons that use sarcasm and irony a little bit too often, but that wants to be straightforward, non-ironic, and frank once in a while.
Inory is an oral and verbal marker used in situations where people misinterpret your honest intention of being nice/direct/straight/frank, and misunderstands it as you being sarcastic or ironic.
It is used to tidy up messy situations where sarcasm (and irony) ain’t that appropriate - but where, you somehow, have managed step right into the bear trap of being seen as some kind of insensitive asshole that blatantly flaunts his/her evil wits whenever he get the chance.
In other words: Inory is when you say exactly what you mean, without trying to be smart, witty, hilarious, or bitter.
X: Look at my new shirt. I bought it yesterday, isn't it nice?
Y: Yeeaah... *hrm* ...rrreally nice...
X: What? What's wrong with it!?! I think it is cute.
Y: ...Oh, nothing. I mean, what could possibly be wrong about a *hrm* _shirt_ like that? ...Really nice shirt; I wished I had one exactly like that, that's how much I like your shirt. I mean: Oh goody! What a shirt!
X: You're mean! Why do you have to be mean and sarcastic all the time?
Y: Hey hey, easy now! I wasn't mean, ironic, or sarcastic at all. Pumpkin, I was just INORIC. I really mean it when I say that I like your new shirt.
X: Oh? Hahaha! Oh so you do like the shirt!
Y: Of course *hrrm* I do. How could anybody not like that shirt...
X: Uh? What did you mean by that? I thought you j-...
Y: Inory baby, inory.