Among all the $20 gold coins issued with the Liberty Head design, the 1871 is ranked 35th in terms of rarity. In overall population, it is scarce, but it is not as rare as the 1859, 1862, 1863, 1868, 1881, 1882, 1885, 1886, and 1892. In Mint State, the 1871 $20 is rare. Akers claimed to have seen a blazing Gem, but neither PCGS nor NGC has graded or certified it yet - the finest examples seen by them top out at MS64.
David Akers (1975/88): The 1871 is one of the rarest P-Mint twenties, distinctly surpassed in overall rarity by only the 1861 Paquet and the seven ultra low mintage issues from 1881 to 1887. (It is also just a little less rare overall than the 1859.) As a date, the 1871 is about on a par rarity-wise with the 1862 and 1868 as well as the 1873 Closed 3 and the popular, low mintage 1891. Most known 1871 twenties grade EF or AU although, of course, some well worn ones are also around. In strictly uncirculated condition the 1871 is rare, a little more so, in fact, than either the 1862 or 1868. I have seen one blazing gem 65+ coin.
Heritage 1/2007:3719, $54,625 - Gilded Age Collection (Robert J. Galiette) - Stack's/Bowers 8/2014:12043, $44,062.50