Art from here!
The Hounds of Halthrag Keep is the first time I've seen roll under and roll over stat checks - you roll over your Strength to see if you can wedge yourself through a crevice, for instance. I thought this was brilliant (thanks !). I wonder if it's possible to have a Black Hack-ish hack where all the stats are explicitly for both rolling under and over.
Murderous is a combat skill, you roll over it if you're trying to blend in with non-adventurers or convince a constable that you're not some wild-eyed murderhobo on a cocaine bender, they went that way, swear on me mum's soul.
Stubborn, you roll over this to assimilate new information that you would normally find odious. New languages, maybe, or to recognize that these orcs aren't trying to kill and eat you. Roll over to understand something that breaks reality, maybe, roll over it to not lose Mental Health Points a la Perdition (out of mental HP, roll on some Mind Melting table).
Quick - roll over this to do something with steady precision. Surgery, lockpicking. For anything where slow is smooth, smooth is fast, I guess. Someone will say 'that's everything' but running needs to be quick.
Rational - roll under for tinkering, making clockwork automatons or the like, debating like a lawyer. Roll over for magic, reasoning with demons, debating from your dumb-ass gut.
Charismatic - oh God I can't get away from the original six stats, I'm a fraud, you knew it all along. Roll under because you can't come up with a better idea than 'I convince the sheriff these aren't stolen banners' (with disadvantage). Roll over to blend into a crowd or hide, or, hell, sneak around in general. Does this overlap with rolling over Quick well I just came up with this stuff. It overlaps with using rational to debate people, but you don't need charisma for debate, especially if you don't care about winning.
With this stuff you probably want some kind of 'WITHSTAND' save stat that you're almost never going to want to roll over (unless you're trying to get mutated in a radiation zone). Or just call it 'Save' and have it be your level plus 2 or the like, and you want to always roll under.
When you level you can probably try to roll over a stat to increase it or roll under.a state to decrease it, your choice. This is that 'improving crappy stats is easier than improving great stats' rule from the Black Hack but you can decrease, say, Rational, if you want to be a better wizard.
Some monsters have a BULK stat they roll under to pick you up and throw you with. They roll over it to get inside a building.
Some animal-intelligence monsters can have an EAT stat. Roll over EAT to stop fighting the PCs and flee when they put up a good fight. Roll under it to eat PCs.
Demons could have stats like CORRUPT to try to seed you with cancerous thoughts or ideas that you have to Stubborn your way through. Or find someone with crap Rational who learned magic good and have them help you.
Hirelings could have a FOLLOW stat to roll under to stay true to their recalcitrant, mercurial bosses, or roll over to take their own initiative and do something wonderful for the PCs. (Or to stab 'em in the back!).
I would give hirelings one other stat - SKILL - and they roll under it to do what you hired them for (soldiers fight, trappers find or create traps, etc). Roll over it to do anything else that's risky, probably with a disadvantage so that you don't hire a shitty scribe to be your private assassin / engineer / starship pilot / magus. Or maybe they just can't roll over it for more than 1 thing a session, I dunno, that might be too much to track.
I think these stats are a bit too generic - I need to work on channeling a darker Thundarr vibe. Carcosa-ish. Numenera but scarier.
CRAVEN (run away or hide, roll under. Roll over to face down horror)
OLD TECH (still make tech vs do magic)
only thing missing - what do you roll to scavenge? Maybe that's a flat 1-in-6 chance per turn.
Corruption happens to those who wander from (slowly failing, class stratified) settlement. When you fail a corruption roll-over your corruption decreases right then. And you may or may not have to roll on a mutation / corruption table. Some of these effects should be good, some should be neutral (but distinctive), some should be at least somewhat negative. Like you can't use your tentacle hand to wield a weapon for now (needs training from a mutant) but you can use it to grapple like crazy well.
There's overlap so maybe you are using your tentacle hand with MURDER or maybe it's CORRUPTION.
As the awesome Andrew Shields pointed out, this is basically an extension of ideas in Lasers and Feelings. He also urged me to use a die other than d20 for this, which seems reasonable.