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