The dihedral group of order 4 acts on the conjugation classes of hyperbolic elements of $PSL_2(\mathbf{Z})$: by passage to the inverse, by conjugation by an element of $GL_2(\mathbf{Z})$ of determinant -1, and by the product of the 2. Via the dictionary (which will be recalled) between closed hyperbolic and geodesic conjugation classes of the modular surface, the same group acts on modular geodesics, which can therefore acquire non-trivial symmetries by being fixed under one of the 3 involutions above. We will characterize these invariant geodesics and we will see that there is a strong link with Gauss' reciprocal, ambiguous and inert quadratic forms. It is a joint work with C.-L. Simon (Rennes)

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