Adjust oven rack to middle position and heat oven to 450 degrees. Line baking sheet with parchment paper. Pulse all-purpose flour, cake flour, sugar, baking powder, salt, baking soda, cream cheese, and butter in food processor until mixture resembles coarse meal. Transfer flour mixture to large bowl. Stir in buttermilk until combined (dough may appear slightly dry).
Turn dough onto lightly floured surface and knead briefly until dough comes together. Roll dough into 8 by 6-inch rectangle, about ¾ inch thick. Cut into twelve 2-inch squares and transfer to prepared baking sheet. Bake until light brown, 12 to 15 minutes. Transfer to wire rack and let cool 5 minutes. Serve warm.
MAKE AHEAD: Unbaked cut biscuits can be refrigerated on baking sheet, covered with plastic wrap, for 1 day. To finish, heat oven to 450 degrees and bake as directed.
Instead of rolling the dough, cutting round biscuits, then rerolling the scraps to cut more, we roll the dough into a rectangle and cut 12 square biscuits.