A Handle on the Situation
You've guessed it: A handlebar mustache.
The Third Policeman by Flann O'Brien
Somewhere in the thick of this novel, a theory is proposed as to how one might actually go about growing bicycle handlebars from one's face. It boils down to just how much said person rides his or her bicycle. I will not spoil the details of the theory; I must leave that for you to enjoy. I will, however, say that everyone in the novel seems to be obsessed with bikery (that is, all things bicycle-related). Sans our main character, and this makes him an enigma to the policemen!
This book is so surreal and inventive, I really wish it were longer. I'm about 4/5 of the way through it as of the moment of this post.
Two policemen, of titular reference, are lead characters in this novel (the third officer is supposedly always undercover and secreted away, but always watching). They bustle about the surreal county with the main character following them, not necessarily by choice. They dazzle and puzzle him with all sorts of strange theorems and devices and creations. They amaze and scare him, alternatively. He's in quite a pickle at the moment, just so you know. But I can't tell you what exactly it is.