We are dealing with domains of the complex plane which are not symmetric in common sense, but support fixed point free antianalytic involutions. They are fundamental domains of different classes of analytic functions and the respective involutions are obtained by composing their canonical projections onto the complex plane with the simplest antianalytic involution of the Riemann sphere. What we obtain are hidden symmetries of the complex plane. The list given here of these domains is far from exhaustive.
References
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https://doi.org/10.1023/a:1006150004533
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Ahlfors, L.V. (1979) Complex Analysis. Mc-Graw-Hill, New York.
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