Bobby Fischer's Rules of Fischerandom Chess
...For a much simpler version of the rules, see Fischer Explains the Rules of Fischer Random. The rules were also copied into Gligoric's Shall We Play Fischerandom Chess?, in the chapter titled 'Fischerandom Rules' (p.86-92). For other copies of the rules, which likely lead to other resources about chess960, see Google: "pieces on their respective back rows receive an identical random shuffle".
Fischer's rules included the following explanation...
White and Black have identical positions. From behind their respective Pawns the opponents' pieces are facing each other directly, symmetrically. Thus for example, if the shuffler places White's back row pieces in the following position: Ra1, Bb1, Kc1, Nd1, Be1, Nf1, Rg1, Qh1, it will place Black's back row Pieces in the following position, Ra8, Bb8, Kc8, Nd8, Be8, Nf8, Rg8, Qh8, etc.
...where the position used in the example is shown in the following diagram.
Start Position 760
Did the position have some special meaning for Fischer or was it also chosen at random?
Later: Gligoric mentions earlier in his book (p.36) that the rules he quotes were formulated in September 1993, almost three years before the press conference announcing them.