-triple lutz-halfloop-triple flip
This has been done. By Evgeny Plyushenko at NHK one year, I think 1998 or 99.
-triple axel-halfloop-triple axel
Why a half loop? That would put the skater on a back inside edge (of the left foot for a counterclockwise jumper). Then they'd probably have to take a couple of steps to get onto the forward outside edge, in ways that would dilute the effect of the sequence. Might as well just do a tap-toe or mazurka in between. (Common with single axels at low levels, I think Surya Bonaly and Jessica Mills used to do this with doubles.)
Or after the half loop, turn a back inside three onto the forward outside takeoff edge. That would be very impressive, but I really doubt it would work with a triple axel. Let's see it with single or double first.
We have seen triple axel-half loop-triple salchow and -triple flip, if that's what you're thinking of, from Yagudin and Plyushenko respectively.
The triple lutz-triple flip is something that I don't think has ever been done because of the fact that to do them in combination, you have to do the second jump in the opposite direction, and I don' t know one skater that can jump in both directions.
You mean triple flip-triple lutz? The problem with putting a flip at the end of the combination is not that you need to reverse direction, but that you need to get onto the back inside edge. So a half loop in between would do it (and has, see above), or intentionally landing the lutz on the "wrong" foot (not likely with a triple), or changing direction with a change of edge to (which, like the half loop, would make it not a true combination).
Anyway, just as toe loops and loops are the jumps that can be put at the end of normal back outside landings for true combinations, and salchows and flips at the end of jumps with back inside landings (one-foot axel, one-foot double salchow, half loop), the obvious second jump in a combination that rotates both directions would be a lutz. Any jump landed on a back outside edge as normal, followed by a lutz in the opposite direction. The most I would expect to see at this point even at the highest levels would be a double jump in the bad direction and a triple in the good direction. E.g., we have already seen reverse walley into normal triple lutz. So maybe start with reverse double toe into triple lutz. Or triple anything into reverse double lutz.
Again, not likely with triple-triple combinations. I've seen it done with singles at low levels, and there are a few skaters who *can* do some triples in both directions but not consistently enough to put them in combination (e.g., Rohene Ward).
What I'd like to see is the code of points add an automatic +1 or +2 to the GOE for a combination or sequence with double or higher jumps in each direction, or a bonus to the base value of the jump performed in the opposite direction or to the value of the combo/sequence as a whole. It would be harder to reward a solo jump done in the skater's bad direction because there wouldn't be any immediate cues within the jump itself that it's an "opposite direction" jump for that skater; it would look just like a normal jump by a skater who jumps the other way. But if the tech specialists can keep track of each skater's normal jump direction and identify the exceptions, then add a bonus to base value or GOE for those too.
Some of gio's suggestions seem like fun ideas. Just need to make sure the system would reward them. Putting walleys and inside axels, at least as doubles, into the scale of values (maybe singles too but specifying that singles of those jumps don't block jump boxes) would encourage people to master the doubles. And someday maybe eventually someone could do a triple. I think the doubles of those jumps need to be worth more than double axels, possibly more than the easier triples.
And I'm still waiting for double one-foot axel into triple salchow.