"Today I received a call from the company representative saying the order was faxed to us by mistake so they were surprised to receive an invoice from us. They intended to send the order to another abstractor and had in fact sent the order to her as well."

To give the benefit of the doubt, they could have attempted to fax it to the other abstractor when it came to you. Upon follow-up with the other company determining the fax was not received, it was then faxed again correctly. This would explain the 'surprise' at receiving the invoice. It did not occur to them the first fax had successfully transmitted to an incorrect party.

"They say I should have noticed their error and called them before proceeding and that it was my error not to do so."

However, they should have investigated when the first fax did not reach the intended recipient. Therefore, there was an error on both sides.
 
A split (or discounted) fee would be the only equitable solution.

If, in fact, they knew the fax had gone to the wrong party, then they should pay the full fee. It was their error for not contacting you when the mistake was realized.

James Powell