Orotine is one of the nine Muses built by Van Rijn for the Storm King. She is the Muse of Geometries; in her own this means she deals with "the shapes of things, and their relationships with each other."
We also don't know what she's been doing for the last two hundred years, but she seems to have weathered the decades better than several of her fellow Muses, and finally appears personally in-comic when she Tarvek at a royal party in England. (This is also the first time the comic has given her name.) She later speculates that she was summoned to the soiree by Queen Albia as a distraction for him away from Agatha, an attempt which fails. Orotine goes on to to Tweedle, whose claim to the Lightning Throne she contests, and then Gil, who to make a better impression, and who asks her for relationship advice vis-a-vis Agatha.the stunned Muse-expert
In a laterwith Krosp, she states that her official ranking of Tweedle's claim to the throne is "a distant third at best". She does not say who is in front of him.
When Albia sends an airship expedition to the Continent to deal with a Polar Ice Lord invasion, headed up by Gil and Tarvek, Orotine accompanies them.
Before appearing in the comic, she was named in The Muses Deck created for the real-life card game Pairs. (In which she again displays the L-shaped whatever-it-is.)
She appears in the prologue of Agatha H. and the Clockwork Princess with the rest of her sisters, belatedly warning the Storm King that his impending bride-to-be intends to betray him. The narrative notes that she had produced maps which proved highly useful when Andronicus was planning his battle strategy.