Your comments

I can think of a couple of other confusing scenarios... for instance, if I reordered items, would the team member get to know about it (if I forget to tell them)? Would team members themselves be able to reorganize the items? I think this really depends on the kind of team you're dealing with. Some may be able to juggle their stuff around more freely, some may have items that must be done but other items they can mix around.

Also, I assume I should not be able to rearrange anything in "done" status. I haven't checked if something can be marked "undone".

One obvious way to solve this is to make the order/reorder function optional and add it as a configuration possibility. But adding configuration options can be a really slippery slope in terms of managing the product and its complexity (complexity drives up testing costs...).