Partiality Continues to be Confusing, part 2

Common Trip-Ups:

Confusing Impartiality with Symmetry. Symmetric positions seem impartial because each of the moves for one player has an opposite in the set of moves for the other player. Those opposing positions have opposite values, however. While it is true that all impartial games are symmetric, the converse does not hold.

Separate Scores. Many games are nearly impartial, except the players keep track of different scores. The game board (not including the scores) may be changeable in the exact same ways by both players, except that then the players get different scores. 3,6,9 and Odd Scoring are good examples of this. These games are not impartial, because the scores are different.

All the positions are impartial games. { 0, *, *2, *3 | 0, * } may look impartial because all the positions are impartial positions. This is not impartial, however, because the players don't have all the same move options.

The Position is equivalent to an impartial game or 0. A game equivalent to * is not necessarily impartial. The game could be: { 0 | 0, *2 }, which is equivalent to * but does not have the same options for both players. Equivalence does not preserve impartiality. (Perhaps not everyone agrees with this!)

What are some other common problems I didn't list here?