ElgShould point out that corks and bios actually depends on where your head is pointing, but for rodeos and misties it matters which way the trick is set. A regular rodeo is set backwards, up hill. A switch rodeo is is still set backwards, but since you're going switch into the jump it is set down hill. When doing corked tricks your head will be pointing up hill both on regular and switch tricks, and when doing bios your head is pointing down hill for both regular and switch.
This is really basic explanations, off course the different axis' vary from trick to trick and skier to skier.
Thanks for the explanation. I know what you mean by backward, it's just ambiguous if people are not familiar with the trick.
So, basically a cork is a head-point-to-up-hill spin, doesn't matter if it includes flips or not. I n this sense, what I described is also a cork. But people would call a real backflip cork(inverted at some point) an underflip.
But bio doesn't include flips, if a frontflip is involved is a misty. if a backflip is added, it's a rodeo.
I think the rodeo you described is not correct. It's set backwards(as a backflip-ish) but has to set side-hill or down-hill, like(https://youtu.be/PVxul3gAGTs?t=23s
)(well, call that trick a flat is fine, they all set the same way). A switch rodeo is also down-hill (or at least sideway, cuz the physical body restriction) (https://youtu.be/O5iaAtNuWuw?t=12s
The reason I bring it up is because, originally I thought a cork is a regular off-axis spin with backflips but not as a inverted backflips. That's why I'm curious a off-axis spin pointing uphill with a frontflip. I haven't seen any video, it's just come from my thoughts.
After the discussion, I believe a cork is just because is head-point-up-hill-off-axis-spin. So doesn't matter if it's pure spin or frontflip or backflip.