The book Symmetry in Chaos by
Michael Field and
Martin Golubitsky
(then both at the University of Houston) was published by Oxford University
Press in 1992. (A paperback edition appeared in 1995 and a second edition, published by SIAM, appeared in
2009.)
The book contains explanations and examples of symmetric chaos.
Some other sources for general
expositions of symmetric chaos include

Symmetric Chaos.Computers in Physics,
September/October, 1990, pages 470-479.

Symmetries on the edge of Chaos.New Scientist,
Vol 137, 9 January, 1993, pages 32-35.

Symmetric Chaos.MATH Horizons,
April 1995, pages 34-35.