What do we know about hexagonal chess variant design?

This thread is intended to find some ideas how to design hexagonal chess variants. The information I could find on these games in the net is limited to descriptions of their rules, so I am writing down some ideas and deductions in advance of a potential chess.com hexagonal chess server.

0) The hexagonal grid does not invalidate any existing advice on designing good chess variants

https://www.chess.com/clubs/forum/view/good-custom-position-bad-custom-position-dos-and-donts-for-ncps-1

https://www.chess.com/clubs/forum/view/variant-design-patterns

https://www.chessvariants.com/opinions.dir/fergus/design.html

1) The hexagonal grid has 12 natural directions

This is a source of much unintentional confusion surrounding the terminology of the rules. Orthogonal pairs of directions running through the edges or corners of the cells do not actually exist unless you use Strozewski’s board with distorted cells in spite of the convention of calling the directions running through the edges of the cells “orthogonal“.

2) Diagonals run through the edges and corners of the cells

This is the reason for coloring the cells three ways. It also clears up the unintentional confusion due to the convention of calling both sets “root-3” diagonals.

3) A royal piece should have moves through all sides of the cells and balanced moves through vertices of the cells if it has any

This undermines the valid possibilities where all root-4 diagonals are this royal piece‘s moves and any edgewise directions used are the Knight analog’s moves.

4) The Rook analog should have moves in directions such that one of it can reach anywhere on the board without a kluge rule and the Knight analog should have all the closest moves not on the mobile royal piece’s lines

Unfortunately, this creates traps for most of the valid unnatural possibilities to fall into.

https://en.wikipedia.org/wiki/Cross_chess Rook analog cannot have both horizontal and vertical moves without a kluge rule unless you use Strozewski’s board with distorted cells

https://en.wikipedia.org/wiki/Masonic_chess, https://en.wikipedia.org/wiki/Masonic_shogi (Bishop analog has only diagonal root-4 moves) Knight analog omits vertical move(s)

5) That said, the valid unnatural possibilities come out like compromises between ancient Chaturanga and modern Chess

This is due to them normally rendering Knight moves as root-4 diagonals.