Considering scalar complex-valued functions $f(\theta)\in\mathbb C$ for $\theta\in[0,2\pi)$ on a circle, what is the name (if there is one) for symmetry after rotating $\theta$ by $\pi$ and conjugation?
\begin{equation} f(θ) = \operatorname{conj}[f(\theta+\pi)] \label{eq1} \qquad\qquad\qquad(1) \end{equation}
Is it a symmetry group with a common name and notation? ("no" is a fine answer too!)
Edit: I ran into this when considering the symmetry of Fourier transforms of 2D real-valued signals, evaluated along a ring in the frequency domain with a constant wavelength. So, it's another flavor of the more familiar conjugate-symmetry for 1D Fourier transforms of real-valued signals. I don't know it's name, though.
Maybe another way to ask this is: What notations are available for talking about the set of functions satisfying $(1)$ abstractly? For recognizing other algebraic structures that may share similar properties?