@YCor: I cannot tell you much about the meaning related to the geometry/properties of the group. I got interested in this question through certain properties of so-called (two-sided) Wiener Amalgam spaces, where the norm of $f$ is essentially the $L^1$ norm of the maximal function $M f(x) = \| f\|_{L^\infty (QxQ)}$. I then wanted to understand the norm of $1_{x Q}$, in particular compared to the norm in the one-sided Amalgam spaces.

This is really incredible! Thanks! I still need to check the fine print and understand parts of the intuitive explanations that you provide, but I could follow the main argument. One thing you did not stress: the uniqueness statement of the Riesz projection theorem is what allows to deduce the conjugation invariance, right?

@ მამუკა ჯიბლაძე: The Heisenberg group is such an example. I can add the details later today if you are interested.

The OP is working on the torus $[-\pi,\pi]$, not on the whole real line.

@LksRheil: If you are interested in the Fourier-analytic part, one reference is "Probability theory" by Bauer. I only have the German version right now, but there the precise statement you want is Theorem 23.4.