Upstairs I found the rest of the set as well as a plain "black & white" (or red and white) version.
Notice that the label calls it an "Alice in Wonderland" set, but the box just refers to it as "The Alice Chess Set". This is probably as it incorporates characters from Through the Looking-Glass (Queen Alice, the white king, the white knight, Tweedledum and Tweedledee) and Alice's Adventures in Wonderland (white rabbit and the mad hatter). The box also features queen Alice and the queen of hearts.
Interestingly, there's a full list of pieces for both sides at the beginning of Through the Looking-Glass, but this doesn't include Alice. Likely the decision was made to have the set feature more well-known characters, such as the ones used in the Disney version (though the white knight is anomalous to that idea).While finding information on the set, I also discovered the rules to Alice chess.
Alice in popular culture sighting no. 39: In the Simpons episode, "The secret war of Lisa Simpson", just after the kids enter the "museum of crime", Chief Wiggum warns them they are about to go "through the looking-glass".
Alice in popular culture sighting no. 40: Once again, on the Art of Neighbours group on Facebook; this picture of Mickey and Ben was posted by Sion Pennant-Williams and Rosie Hedger referred to them as "tweedle dum and tweedle dee".
