  Rohatgi and the author recently proved a shuffling theorem for lozenge
tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be
considered as a hybrid between two classical theorems in the enumeration of
tilings: MacMahon's theorem about centrally symmetric hexagons and
Cohn-Larsen-Prop's theorem about semihexagons with dents. In this paper, we
consider a similar shuffling theorem for the centrally symmetric tilings of the
doubly-dented hexagons. Our theorem also implies a conjecture posed by the
author in arXiv:1803.02792 about the enumeration of centrally symmetric tilings
of hexagons with three arrays of triangular holes. This enumeration, in turn,
can be considered as a common generalization of (a tiling-equivalent version
of) Stanley's enumeration of self-complementary plane partitions and Ciucu's
work on symmetries of the shamrock structure. Moreover, our enumeration also
confirms a recent conjecture posed by Ciucu in arXiv:1906.02951.
