Chess in a Klein Bottle

After the Chess in a Moebius Strip , what is the next canonical step?

Take the Moebius Variant, and join columns a and h . This creates a Klein Bottle. This can't be done in our three-dimensional space! The board becomes like this:

	       Black
    a   b   c   d   e   f   g   h
  +---+---+---+---+---+---+---+---+
14| P | P | P | P | P | P | P | P | 0
  +---+---+---+---+---+---+---+---+
13|   |   |   |   |   |   |   |   |-1        The 15th line is identical
  +---+---+---+---+---+---+---+---+          to the 1st, in other words,
12|   |   |   |   |   |   |   |   |-2        the 9th line is the -5th.
  +***+***+***+***+***+***+***+***+
11|   |   |   |   |   |   |   |   |-3        White's pawns in the 2nd line
  +---+---+---+---+---+---+---+---+          or Black's pawns in the 9th
10|   |   |   |   |   |   |   |   |-4        (or -5th) move forward
  +---+---+---+---+---+---+---+---+
 9| p | p | p | p | p | p | p | p |-5        White's pawns in the 0th
  +---+---+---+---+---+---+---+---+          (14th) or Black's pawns in
 8| r | n | b | q | k | b | n | r | 8        the 7th line move backwards
  +---+---+---+---+---+---+---+---+
 7| p | p | p | p | p | p | p | p | 7        The 11th/12th (or -3rd/-2nd)
  +---+---+---+---+---+---+---+---+          junction is done after a
 6|   |   |   |   |   |   |   |   | 6        rotation of the board, so
  +---+---+---+---+---+---+---+---+          that the a11 is in front of
 5|   |   |   |   |   |   |   |   | 5        h12, and so on
  +---+---+---+---+---+---+---+---+   
 4|   |   |   |   |   |   |   |   | 4        Each line rolls around
  +---+---+---+---+---+---+---+---+          itself, so that a White
 3|   |   |   |   |   |   |   |   | 3        Pawn at a11 is attacking
  +---+---+---+---+---+---+---+---+          h12
 2| P | P | P | P | P | P | P | P | 2
  +---+---+---+---+---+---+---+---+   
 1| R | N | B | Q | K | B | N | R | 1
  +---+---+---+---+---+---+---+---+
    a   b   c   d   e   f   g   h
	     White

All other rules (castling, en-passant, promotion, etc) apply; contrary to the Torus Chess , where all columns are equivalent, the torsion in the board create a kind of non-symmetry: the a-h column is equivalent to the d-e column, but they are different to the b-g and c-f columns

Like in the Moebius Chess , along the 8-9-10-11-12-13-14-15 board, each King is facing the opponent's Queen; and the Bishops don't have a fixed colour

Since in an empty board in the Torus Chess a Bishop may reach, in one movement, all squares of the same colour, it's expected that in the Klein Chess a Bishop might reach all squares. Let's check this property: a Bishop starting at f1 might move in the diagonal f1- a6- h7- d11- f12- h14- a1- h8- a9- c11- e12 -c14 -b1 -a2-... -e1 -h4 ... -f1 -a6; or it might move in the other diagonal f1 -h3 ... f1. So, the Bishop can only reach 3/8 of the Board in one movement

This variant can lead to subvariants, or similar games:

Torus Chess , if there is no twist between the 11th/12th lines
Moebius Chess , if the a and h columns are separated

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