Many people seeing this word would at once recall Lewis Carroll’s poem Jabberwocky from Through the Looking-Glass: “‘Twas brillig, and the slithy toves / Did gyre and gimble in the wabe.”
You might enjoy the version created some years ago by the British satirical columnist Miles Kington in the style of Raymond Chandler:
Outside in the street, the first lights had come on and the slithy toves were doing whatever they do in the wabe. Some days they gyre, some days they gimble. It’s no skin off my nose, but I wish they’d make their minds up, then we could all rest easy.
When the toves gyre they spin around, revolve or whirl, an animal impersonation of a whirling dervish. You might link it to gyrate or gyroscope, which would be appropriate, since all three words are from the same source, the Greek guros, a ring or circle. As a noun gyre means a spiral or vortex. Geographers use it for a circular pattern of currents in an ocean basin, such as the North Pacific gyre, which is infamous as a perennially rotating mass of unrottable plastic rubbish. Like gyrate and gyroscope, gyre is said with a soft g, except in Jabberwocky, in which Lewis Carroll insisted that it should have a hard g.
Another famous appearance:
Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold.
The Second Coming, by W B Yeats, 1919.
No one, by the way, is sure what slithy toves do when they gimble. It was one of Carroll’s lesser linguistic inventions and hasn’t caught on. Humpty Dumpty, Carroll's alter ego, suggested that they were making holes like a gimlet with their corkscrew noses. Carroll might also have had gambol in mind, or perhaps gimbal, a contrivance for keeping an instrument such as a compass horizontal in a moving vessel. If so, pace Miles Kington, toves must simultaneously gyre and gimble, spinning to stay balanced.