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Great, fast, specific, usable response.  I was aware of some, but you
saved me time and helped me focus on exactly where I should buy for
confidence.
Thanks much, WEB7

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Super; thanks!  From the first link you provided I found
http://www.geocities.com/rodeodrive/9830/ which is someone's attempt
to build a census of all rotated die coins.  My 1861 three-cent silver
is definitely among the 180-degree rotated set.  According to the
site, there's at least 30 other such 1861 three-cent silver coins as
well (the one pictured in the link you found being one in particular).
 That pretty much answers my question.  Thanks!

thank you playhosea for the great answer!  i finished looking though
all the links and had a really fun time of it.  thank you for such a
thoughtful answer and for taking the time to comment on each of your
choices.  as it turns out, there wasnt a single choice of yours that i
did not find interesting.  thanks for the good leads!

my only regret is that more people did not comment.

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Thank you so much for your help!!!