That's what I'm beginning to understand regarding the mounts.
So say you have one of those transition-era blades. Its the proper length where it could really be a tachi or katana (disregarding the side the mei in on). If the owner mounted it in tachi-koshirae its now a tachi and if its in katana-koshirae, it's now a katana. And as I've been learning more and more, almost no original mounts seem to exist from the 1400-1500s and most koshirae for blades these days seem to be modern make or edo period mounts.
I'd imagine if a wealthy swordsman of the time asked a swordsmith to sign a blade on "the other side" for whatever reason, they'd probably do it. We're all human after all and history is full of outliers even though I've started to become fascinated by the attributions that the experts in nihonto can make because the Japanese did seem to standardize quite a bit making this whole system of grading and attribution to smiths/eras/schools/etc. possible. It's all becoming more and more fascinating to me the more I learn.
If I saw that Sunnobi Tanto you posted without any classification, I'd say its a wak but again, I know very very little (and even less and less about tanto). I've gotten through Nagayama's book briefly and still, 99% of the stuff probably hasn't stuck but I'll be attending the Las Vegas show with book in hand and hopefully some swords in hand with some people who can educate me in person!