In a medium bowl, whisk together flour, cornmeal, baking powder, and salt. With an electric mixer, cream butter and sugar until light and fluffy. Beat in vanilla, egg, and orange zest. With mixer on low, gradually add flour mixture; beat until just combined. Stir in cherries.
Transfer dough to a lightly floured surface; pat into a log, 8 inches long and 2 inches wide. Wrap dough in parchment; form it into a rectangle by flattening the top and sides with your hands. Twist ends of parchment to seal. Refrigerate until firm, at least 1 hour.
Preheat oven to 350 degrees. Line two baking sheets with parchment. Slice dough 1/4 inch thick, rotating log 1/4 turn after each slice. Place 1/2 inch apart on baking sheets. Bake, rotating sheets halfway through, until cookies are firm to the touch, 10 to 12 minutes. Cool cookies on a wire rack.
Be sure to rotate the log a quarter turn after making each slice. This will help it maintain its shape.