Actually, the game can save changes, but only if you change the right number of things. Changing appearance only is impossible it seems, but if you change powers as well it seems to save the changes. I'm not sure of the exact combinations of things that you will need to do, but changing
everything covers it.I'm guessing that when they were testing, EA made changes to each aspect of the CAH, saw it worked, thought, "OK, that's fine," and didn't go through it trying various combinations of changes.
If you only want to change one aspect of your hero, here's my advice: take note of the things you want to change. Then go through and change everything - appearance, attributes, and powers - to anything, not what you actually want. Make at least one change to each. Click on a different hero, then go back to the original. Take a moment to catch your breath. Make the changes to the hero you actually want (and restore the things you didn't want to change in the first place), and hey presto, everything should be saved.
I may take a closer look to see if there is a quicker way some time.
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