Corner Orientation, abbrevaited CO, the orientation of a cube's corners. There are three possible corner cubie orientations. CO is a sub step in many methods.
For the last layer of the 3x3x3 cube there are three main variations used for the seven cases of CO, one that is named 'pure CO' wich orients the corners without affecting anything else, not even corner permutation, one preserves edge orientation wich is OLL with edges correctly oriented and the last variation orients the corners ignoring edges.
For last layer corner orientation preserving edge orientation but ignoring permutations, see OLL 21  27.
See also:
Pure CO
All cases here have long algorithms, the H case is actually the worst LL case of them all and the pi case is second worst. The good thing is that all cases are solveable using only two sides (RU 2gen). The first alg given for each case is optimal in 2gen and the second is optimal in Half Turn Metric, using as many sides that is needed for that (26).
Two corners unoriented
Three corners unoriented
S

S

Four corners unoriented
H

pi
