Exhaustive search shows that there are several ways to fill all faces
of Rubik's 3^3 with Latin squares, but none lie in the primary orbit.

Here are two arrangements in the orbit where one corner is twisted
1/3 turn anticlockwise:

UFB                           UBF        
BUF                           FUB        
FBU                           BFU        

LUD RLF RUD LRB               LDU RLF RDU LRB
DLU LFR DRU RBL               ULD LFR URD RBL
UDL FRL UDR BLR               DUL FRL DUR BLR

DBF                           DFB        
FDB                           BDF        
BFD                           FBD        

I did the search with pencil and scissors on quadrille lined paper.
The following observations speed the search:
1) The 3*3 Latin square whether reduced or not must be some
rotation or relabeling of the following pattern: ABC
BCA
CAB
2) The diagonal bars of the squares must be arranged as in
the pretty pattern called "Laughter" because of the shape
of the corner cubies. (See the bars in the patterns above.)
3) When you attempt to place one of the remaining four corner
cubies, the corner color propagates to two edges which restricts
the other color on those edge cubies to not be that color and also
not that color's complement (e.g. U and D). This restriction
then propagates to another corner.

-Roger Frye