Hi friends,

So which do we have?

Corner permutations:
R'FR'B2RF'R'B2R2 (basic + inverse)
(L2 U) (B2 U')*2 (L2 U) (B2 U B2 U')
[RB'R'B]x3 (13)(27)
UL2UR2 U'L2UR2 U2 (Ron)
U'R2UR2UF2 U'R2U'R2UF2 (Ron)

Edge permutations:
RU'RURURU'R'U'R2 (+ inverse, mirrored F/B, inverted mirrored F/B. These
algorithms are really great!!)
MD2M'D2 (in any direction)
M'UMU2M'UM (in any direction, useful if you can do an easy F/B setup move
that flips edges!!)

Edge orientations:
ERERERER'ERERERER' (in any direction)
(M'U)*4 (UL, UB, DF, and DB)
(M'U')*4 (UR, UB, DF, and DB)
(M'U)*4 (MU)*4
(MD')*4 (M'D')*4
R2 D' (R2M2) (M'U)*4 (R2M2) D R2
F D F D' (EF')*4 D F' D' F'
M' U M' U M' U2 M U M U M U2
(DwDRwR)*3 (octaflip)

Corner orientations:
R'DRFDF' Ux FD'F'R'D'R Ux'
(R'U2RUR'UR) U2 (L2U'F'BL2FB'U'L2)
(R'U2RUR'U'RUR'UR) (F2U'LR'F2RL'U'F2)
(LU2L'U'LU'L') (R'U2RUR'UR)
(F2 L F2 L') (U2 R U' R' F2 R' F2) (R U')
(F' D2 F R' U2 R) * 2

Basically you can do any two generator corner orientation (like RU2R'U'RU'R'
or RU2R2U'R2U'R2U2R), then check which edges are cycled, then undo the edge
cycle (see Edge permutations).
For the corner orientations there are many neat tricks in case the total
orientation of LL is not 0.

What could we add to this list?

One idea I got from blindfold solving to transfer to speedcubing is this:
suppose you could do the orientations (edges/corners) both in one step, and
the permutations (edges/corners) both in two steps. Then in 15 seconds
preinspection you could memorize the orientations. After that you do the
permutations. Total of 6 steps. That is better than the cross/F2L/OLL/PLL
system with 7 steps.
Another thing is this: without looking at the cross, just orient all edges.
This is much easier than for blindfold because you can do F to flip 4 edges.
I think the maximum depth for edge orientation is 7 moves. OK, now
preinspect the cube to solve the cross using moves in the group
(U,D,R,L,F2,B2). This would leave you a very easy F2L using only U, R and L
moves. And it would leave you a LL with oriented edges. Try this a few times
and then take an average not including the first step of orienting the
edges. Mine was better than my normal average.
Now if you could orient the edges very fast, and still be able to afterwards
solve the cross fluently, then you could maybe even improve your times.
This system can also be useful for one-handed cubing (many easy moves!!!)
and for beginners who have trouble with OLL or F2L (only 8 corners case for
OLL, no rotations or really hard cases for F2L).

Have fun,

Ron