Are there combinatorial games of finite order different from $1$ or $2$? Are there any combinatorial games whose order (in the usual addition of combinatorial games) is finite but neither $1$ nor $2$? Finding examples of games of order $2$ is easy (for example any impartial game), but I have not been able to think up an example with finite order where the order did not come from some sort of symmetry (for example even though Domineering is not impartial, it is easy to see that any square board will give a game of order $1$ or $2$), and such a symmetry only gives $1$ or $2$ as the possible orders.

There are games of order $4$ such as $$A=\\{1|0\\}+\\{*|-1\\}$$ since $A+A=*$ and so $A+A+A+A=0$.