In a large mixing bowl, beat butter and shortening with an electric mixer on medium to high speed for 30 seconds. Add sugar; beat until combined, scraping sides of bowl occasionally. Beat in almond extract and salt until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour; continue stirring until dough comes together, or use your hands to knead dough, if necessary.
Shape dough into 1-inch balls. Place balls about 1 inch apart on ungreased cookie sheets. Press your thumb into the center of each ball. Drain maraschino cherries. Pat cherries dry with paper towels. Cut each cherry in half. Place a cherry half, cut side down, in the indentation of each cookie.
Bake in a 350 degree F oven for 12 to 14 minutes or until bottoms are lightly browned. Cool on cookie sheet 2 minutes. Transfer cookies to wire racks and cool completely. Makes about 48 cookies.