The Five Game
The reason I call this "the five game" is that it is a game in which five different moves are derived from a five by five grid. This leads to five kinds of pieces.
Given the importance of the number five to the Chinese and some remarkable similarities to Xiangqi, it is probable that the game is of Chinese origin or was adopted in China very early.
In later games, in particular European (Italian, Modern, Etc.) Chess, only the knight's move remains unchanged. Nevertheless, in less derived games, including those speculated to be the earliest (Shatranj, Chaturanga), three of the five are intact. Xiangqi notably retains four of the five (and the cannon has tantalizing hints of the fifth).
While the simple move diagram shown below explains much, it also leaves much unanswered. Given five piece types, why are there eight pieces total? Do the eight pieces derive from the eight by eight board? Or does the board derive from the number of pieces?
For whatever reason, the two stepping pieces were only used once, whereas the three jumping pieces where each used twice. Also missing are the pawns? So very different, but paired, file by file in two neat rows. My personal theory is that they derived from another game; possibly a Greek wargame that reached Asia during the Greco-Buddhist period. (See Rooks and Pawns.)
There's no real way to know if rotational or mirror symmetry was used. Descendant games have a mix.