In a large mixer bowl beat shortening and butter for 30 seconds or until softened. Add sugars, pie spice, baking powder, baking soda, and salt. Beat until fluffy. Add eggs and vanilla; beat thoroughly. Beat in flour. Stir in oats, cherries or raisins, and orange peel.
Generously fill a 1/3-cup dry measure with dough and drop onto a greased cookie sheet. Press into a 4-inch circle. Repeat with remaining dough, placing cookies 3 inches apart on greased cookie sheets.
Bake in a 375 degree F oven for 8 to 10 minutes or until edges are golden. Let stand for 1 minute. Transfer cookies to a wire rack to cool.
Store cookies in an airtight storage container at room temperature for up to 3 days; or freeze, in a freezer container, for up to 3 months. Makes about 14 large cookies.