It references the British expression 'Stiff upper lip'.
It was introduced by Gracie Allen in the 1937 film A Damsel in Distress.
The song is the occasion for an elaborate dance routine performed inside a funhouse by Fred Astaire, George Burns, and Miss Allen, and makes full use of all the trappings of the house, including the funhouse mirror.
[1] The number won an Academy Award for Best Dance Direction.
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