Here's one way of thinking about the monodromy representation associated to the irreducible triple cover of S1.
disclaimer
(You may have to rotate the picture a bit -- just click and drag -- until you can see what's happening.)

Abstractly, we know there ought to be a group homomorphism

&pi1(S,s)-> Aut(Ts)
The fundamental group of the circle is isomorphic to Z. In fact, let g denote the loop corresponding to ``walk counterclockwise once around the circle''; it's a generator of &pi1(S,s).

The fiber Ts consists of three points, which we'll arbitrarily number. In this way, what we really have is a representation

Z -> Sym( 1 2 3 )
Dragging the slider up the screen shows that going around the circle cyclically permutes the fibers; under this homomorphism
g -> (1 2 3)
and the image of the monodromy representation is the cyclic subgroup {(1 2 3), (1 3 2), id } of Sym( 1 2 3 ).

The observation that going around the circle three times returns to the initial state reflects the fact that the image of g has order three.

This is the picture behind some of my mathematics.

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