Thanks to new information coming to light, we have updated the 1956 Hugo Award history page with the finalists that appeared on the ballot that year. We thank Olav Rokne for bringing to our attention an article on page 15 of the 1956 Worldcon Progress Report 3 that included the names of the finalists along with voting instructions.
Note that the order in which the finalists are listed is the same order that they appeared in the progress report and does not imply order of finish on final ballot. According to the article, the final ballot included space for write-in candidates. In Best Professional Magazine, no finalists were listed at all, so all votes were write-ins.
Voting rules in 1956 were significantly different from those currently in use. The shortlist was not formed based on an initial poll as is done today, but apparently by the convention committee. Should we obtain additional information about the detailed voting for this or any other year, we will add it to the relevant history page.