Twitch,
Sorry I did not get on the assessment of your' coin earlier! Kinda been sidetracked with eBay Auctions, going to Estate Sales and looking at Estate Auction items online as well as other Threads here on Treasure Net. Your' coin appears to have a Repunched Date but it does not appear to be any of the 4 known Repunched Date Varieties. There is only one Overdate for the 1881-P $10 Gold coins and it is an 1881 over an 1880 Date. It is possible that your' coin is an Overdate but it would require in-depth analysis of each digit of the Date and the areas surrounding them to see if there is left over evidence of the former digits. All of the Varieties carry a premium over the one's with normal Dates in the same grade and the Overdate is more sought after and valuable. I am leaning toward your's being a Repunched Date but even if it is an Overdate, it is a new Variety which needs to be attributed and recorded. I recommend sending the coin to a Variety Coin Attributer with Coneca to start with identifying what you have and then it will be recorded in the records. Once the coin is returned and you have the Letter of Attribution from Coneca stating which Variety it is or that it is a new Variety which should include a designation, then if you wish, you can send the coin and a copy of the Letter of Attribution to PCGS or NGC, have the coin certified...graded and have not only the grade noted on the holder's label but can also have the (Attribution) Variety noted on the label as well. I think it only costs an additional $18 to have the Attribution added to the label. See the link below for Attribution.
Congrats on obtaining a nice Variety coin!
http://conecaonline.org/content/attribution_services.html
Frank