How many unique ways are there to unfold a cube? By unique, we mean that each pattern unfolded cannot be rotated or reflected to get another pattern.

If you look at the corner of a cube, you can unfold it in (at least) three different ways to get the following pattern:

Thus, to go from one pattern configuration to another, all one has to do is to slide a side from one position to another, while keeping at least one side connected.

__Note__: In actuality, for the proper sides to match correctly, we
wouldn't really be sliding the square, but pivoting it at a right-angle around a
corner.

Start with the simplest pattern of all, by unfolding two opposite sides of a cube and unrolling the rest to get a T-shaped pattern. From there on, we slide any side across another to get almost all other patterns for a total of 10.

However, the last pattern (grey) is unobtainable through this approach for a total of 11. Thanks to Dirk Bell for pointing that out.

__Note__: For the 10 patterns obtained by sliding, the width is equal to
3 squares and the height is equal to 4. For the grey pattern, the width is
equal to 2 squares and the height is equal to 5. The six squares are left (L),
right (R), front (F), back (B), and top (T). The bottom square is left blank.

L | B | R | L | B | L | B | L | B | ||||||

↔ | R | ↔ | ↔ | |||||||||||

F | F | F | R | F | ||||||||||

T | T | T | T | R | ||||||||||

↕ | ↕ | |||||||||||||

B | B | L | B | |||||||||||

L | R | ↔ | L | |||||||||||

F | F | R | F | R | ||||||||||

T | T | T | ||||||||||||

T | ↕ | ↕ | ||||||||||||

B | B | B | B | |||||||||||

L | L | R | ↔ | L | ↔ | L | ||||||||

F | F | F | R | F | R | |||||||||

R | T | T | T |