If the buyer bought the item without it being 
insured, I would have to say that the buyer had 
to be prepared for not getting refunded.  Of 
course, it would have been very very wise for 
you to make that explicitly clear to them before 
they paid for the item, in order to cover your 
back.  A seller usually posts a statement right 
at the beginning that they cannot be responsible 
for the status of an item once it is mailed 
unless the package is insured.

If they paid via credit card, you should ask 
them to check with their provider in order to 
recover their money through their buyer 
protection services.  If the buyer didn't pay 
for insurance (or PayPal buyer protection), you 
have the right to refuse a refund.  (Sure, you 
might look like the bad guy, but there are few 
options here for you anyway.)

As for the allegations that you sent an already 
damaged item, there's not much you can do now to 
convincingly dispute that claim.  If you had a 
previous scan or photo of the item before it 
left you, ask the buyer to provide you with a 
scan of the item from the same angle so that you 
can compare.  If the buyer bought the item 
without even knowing what it looks like, 
unfortunately that's a risk that they were 
supposed to be accepting.

Admittedly, I have been on the flip side of this 
situation where a seller sent me shoddy stuff 
that was incongruent with their item/auction 
description.  And if the buyer here really was 
innocent, I can empathize with them.  As a 
buyer, I tend to ask sellers questions about 
anything that wasn't covered or was unclear in 
their item descriptions.  I would save the 
replies from the sellers as my evidence in case 
the items I did receive turned out falsely 
advertised.  Nevertheless, I've been burned a 
few times before and there has been little I 
could do about it in most cases.

As for the buyer shipping the damaged item back 
to you for a refund, that is your judgment 
call.  Especially if the buyer didn't pay for 
insurance or buyer protection (to which they 
should turn to the policy-holders for a refund, 
not you), you probably shouldn't feel obligated 
to give them a full refund.  If you ultimately 
decide that a refund is in order, more likely 
you would give them your personal value of the 
item in its present state.

As for the buyer keeping the item and asking for 
a partial refund, you need better evidence that 
what has been provided.  Several detailed 
scans/photos, a picture of the actual packaging 
when it arrived to them, etc.

Helpful info?