If you can use the definite or indefinite articles with it, it's a noun.
It refers to the flattening of all pretty people features into a
single, cyborgian look -- a swirling of Hollywood archetypes,
social media beauty ideals, and facial filters. (Business
Insider)
The word " whirl " is often associated with rapid, spinning movements,
as in the motion of a spinning top or the swirling of water.
(Forbes)
Nouns also inflect for number, which swirling does.
At times she paints a little like Van Gogh -- great swirlings of
cloud, as in one of the Dutchman's late, mystical visions of the night
sky (Independent)
Here a whirlpool boils with mud and immense
swirlings of water,
spouting up the slimy sand of Cocytos. (Theoi)
Nouns also have genitive forms, which, though rare, swirling does indeed have.
Intermittently step back and evaluate the vortex effect’s progress.
Make adjustments as required to refine the swirling’s shape,
intensity, and heading. (ITS)
According to these common tests for nouns, there is in fact a noun in use with the spelling swirling.