Invariant tests for symmetry about an unspecified point based on the empirical characteristic function

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Recent and classical tests for exponentiality: a partial review with comparisons (2005, background): “More recent work includes, among others, Henze et al. (2003), Gürtler and Henze (2000), Koutrouvellis and Meintanis (1999) and Kankainen and Usakakov (1998).”
The limit distribution of weighted $$L^2$$L2-goodness-of-fit statistics under fixed alternatives, with applications (2017, background): “Weighted L2-statistics, however, have also been studied for testing non-parametric hypotheses, see e.g. Henze et al. (2003), Ngatchou-Wandji (2009), and Leucht (2012) in the context of testing for reflected symmetry about an unspecified point.”
Testing exponentiality based on characterizations of the exponential distribution (2011, background): “[7], Zhang and Cheng [8], Meintanis [9], Steele and Chaseling [10], Meintanis [11], Jager and Wellner [12], Meintanis [13], Raschke [14], Meintanis [15], Zhao et al.”