I hashed my bag and in it I put... "Is there a hash for unordered items?"
You could use a hash tree for this purpose. It wouldn't have the properties of your "hash", exactly, but it would work for the game.

Use a secret key to derive individual keys for the leaves, e.g. $k_i = H(k, i)$. Have each leaf be a keyed hash of an item in the bag, $h_i = H(k_i, v_i)$.

Then produce a normal hash tree by hashing two leaf values, then pairs of those values, etc. until you get the root, which you publish. You could add any number of dummy leaves (random values) and make the leaves exist at different depths to hide the number of items in the bag.

To show something was in your bag, you'd reveal the leaf key for that item, as well as all the internal sibling hashes up to the root. That would not reveal any other items, since their leaf keys would be secret.