Imagine you have a 3x3x3 picture cube with a different image on each side, take it apart, and then put all 8 corner pieces back in the *solved* state, but put all 12 edge pieces back in a *random* fashion, i.e., in random edge positions and orientations, so that only 1 in 4 of these semi-random configurations can be solved...

Now, in order to ensure that ALL of these configurations can be solved, how many edge pieces must be identical copies? In other words, what is the minimum (necessary and sufficient) number of identical (and therefore exchangeable) pairs of edge pieces required to ensure that every random configuration of a 3x3x3 puzzle can be solved, given that all the corner pieces start in the solved state?

And what is the minimum number of symmetrical (i.e., rotatable) edge pieces (with the same image fragment on both faces) required to solve all of these semi-random configurations, again assuming that all the corner pieces start in the solved state?

Or do you need at least one of each, i.e., at least one symmetrical edge piece AND at least one pair of identical but asymmetric edge pieces?

1 pair needs to be the same for permutations.

And both sides need to be the same on any 1 piece for orientation.

Yes, that makes a lot of sense, thanks.

But what about 2 pairs of identical edges and no symmetrical edges, or 2 symmetrical edges and no pairs of identical edges? Or what about 3 identical edges (i.e., a triplet) with no symmetrical edges? Would these work? Do you really need at least one of each type to ensure that the puzzle is always solveable, or can they substitute for each other?

All edge-orientations are visible => NOT always solvable.
All edges different => NOT always solvable.
All other cases => Always solvable.

Very nice and clear, thank you!