Hmm, other than filtering products displayed by categories, manufacturers, published status etc, you can sort them by almost any column available. Maybe it's just me, but I can't think of what those arrows would do better there, not to mention that it'd be a hell of a lot of work to only change the backend display order with arrows, at least if you have many products displayed.
With categories it's a completely different game as they are typically displayed in a particular order in the frontend, which the backend ordering is mandatory for.
not sure if this helps, but have your customer thinking about it again: why would he want the arrows there?